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  <h1 id="数学-高等数学" class="content-subhead">数学-高等数学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学"></span>
  </p>
  <h1 id="_1">第一部分 高等数学</h1>
<h2 id="1">第1讲 高数预备知识</h2>
<h3 id="1-a_1-dd-neq-0">1. 等差数列（首项  <script type="math/tex"> a_1 </script> ，公差  <script type="math/tex"> d(d \neq 0) </script>  ）</h3>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1 + (n-1)d
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = \cfrac{n(a_1+a_n)}{2}
</script>
</p>
<h3 id="2-a_1-rr-neq-0">2. 等比数列（首项  <script type="math/tex"> a_1 </script> ，公比  <script type="math/tex"> r(r \neq 0) </script> ）</h3>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1r^{(n-1)}
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = 
\begin{cases}
na_1, & \text{r = 1} \\[2ex]
\cfrac{a_1(1-r^n)}{1-r}, & r \neq 1
\end{cases}
</script>
</p>
<p>常用  <script type="math/tex"> 1 + r + r^2 + \cdots + r^{n-1} = \cfrac{1 - r^n}{1 - r} </script>
</p>
<h3 id="3">3. 三角函数基本关系</h3>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/三角函数.jpg" alt="sin_cos" style="zoom:33%;" /></p>
<h4 id="1_1">1）倍角公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin2\alpha &= 2\sin\alpha *\cos\alpha \\[2ex]
\cos2\alpha &=\cos^2\alpha -\sin^2\alpha \\ 
\quad &= 1 - 2\sin^2\alpha \\ 
\quad &= 2\cos^2\alpha - 1 \\[2ex]
\tan2\alpha &= \cfrac{2 \tan\alpha}{1 - \tan^2\alpha}
\end{split}\end{equation}
</script>
</p>
<h4 id="2">2）和差公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin(\alpha \pm \beta) &=\sin\alpha\cos\beta \pm\cos\alpha\sin\beta \\
\sin(\alpha \pm \beta) &=\cos\alpha\cos\beta \mp\sin\alpha\sin\beta \\
\tan(\alpha \pm \beta) &= \cfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}
\end{split}\end{equation}
</script>
</p>
<h4 id="3_1">3）积化和差公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha\cos\beta &= \frac{1}{2}\bigg[\sin(\alpha + \beta) +\sin(\alpha - \beta)\bigg] \\
\cos\alpha\sin\beta &= \frac{1}{2}\bigg[\sin(\alpha + \beta) -\sin(\alpha - \beta)\bigg] \\
\sin\alpha\sin\beta &= \frac{1}{2}\bigg[\cos(\alpha + \beta) +\cos(\alpha - \beta)\bigg] \\
\cos\alpha\cos\beta &= \frac{1}{2}\bigg[\cos(\alpha - \beta) -\cos(\alpha + \beta)\bigg]
\end{split}\end{equation}
</script>
</p>
<h4 id="4">4）和差化积公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha +\sin\beta &= \quad 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\sin\alpha -\sin\beta &= \quad 2\sin\frac{\alpha - \beta}{2}\cos\frac{\alpha + \beta}{2} \\
\cos\alpha +\cos\beta &= \quad 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\cos\alpha -\cos\beta &= -\ 2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}
\end{split}\end{equation}
</script>
</p>
<h3 id="4_1">4. 因式子分解公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
(a + b)^n &= C_n^0a^n + C_n^1a^{n-1}b + ... + C_n^nb^n \\[2ex]
a^n - b^n &= (a - b)(a^n + a^{n-1}b + \cdots + ab^{n-1} + b^n) \\
\end{split}\end{equation}
</script>
</p>
<h3 id="5">5. 常用不等式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\bigg\vert\vert a \vert - \vert b \vert\bigg\vert &\le \vert a \pm b \vert \le \vert a \vert + \vert b \vert \\[2ex]
\cfrac{2}{\frac{1}{a}+\frac{1}{b}} \le \sqrt{ab} &\le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} \ (a,b \gt 0) \\[2ex]
\cfrac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \le\sqrt[3]{abc} &\le \frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \ (a,b,c \gt 0) \\[2ex]
\sin x &\lt x \lt \tan x \ (0 \lt x \lt \frac{\pi}{2})
\end{split}\end{equation}
</script>
</p>
<h2 id="2_1">第2讲 函数极限</h2>
<h3 id="0">0. 基本极限公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to\infty}(1+\cfrac{1}{x})^x &= e ⟺ \lim_{x\to0}(1+x)^{\frac{1}{x}} = e \\
\lim_{x\to\infty}(1+\cfrac{a}{x})^{bx} &= e^{ab}
\end{split}\end{equation}
</script>
</p>
<h3 id="1_2">1. 洛必达法则</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to·}\cfrac{f(x)}{g(x)} &= \lim_{x\to·}\cfrac{f'(x)}{g'(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{a}^{x}f(t)dt}{\int_{a}^{x}g(t)dt} &= \lim_{x\to·}\cfrac{f(x)}{g(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{\psi(x)}^{\varphi(x)}f(t)dt}{\int_{\psi(x)}^{\varphi(x)}g(t)dt} &= \lim_{x\to·}\cfrac{f[\varphi(x)]\varphi'(x)-f[\psi(x)]\psi'(x)}{g[\varphi(x)]\varphi'(x)-g[\psi(x)]\psi'(x)}
\end{split}\end{equation}
</script>
</p>
<h3 id="2_2">2. 泰勒公式</h3>
<video style="border: 1px solid rgba(0, 0, 0, 1);" controls="controls" width="100%" src="/post/数学-高等数学.assets/泰勒级数.mov"></video>

<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n \ \ \ \ \  (令x_0=0)\\[2ex]
&= \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n
\end{split}\end{equation}
</script>
</p>
<h4 id="1_3">（1）常用泰勒级数展开式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
   \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n}}{(2n)!} \\[1ex]
  \cosh x &= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n}}{(2n)!} \\[1ex]
\arccos x &= \cfrac{\pi}{2} - \arcsin x \\[2em]
   \sin x &= x - \frac{x^3}{3!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
  \sinh x &= x + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
\arcsin x &= x + \frac{x^3}{6} + o(x^3) \\[2em]
   \tan x &= x + \frac{x^3}{3} + o(x^3) \\[1ex]
\arctan x &= x - \frac{x^3}{3} + o(x^3) \\[2em]
 \ln(1+x) &= x - \frac{x^2}{2} + \frac{x^3}{3} ... = \sum_{n=1}^\infty(-1)^{n-1}\cfrac{x^{n}}{n} &-1\lt x\le 1 \\[1ex]
      e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{n}}{n!} \\[2ex]
  (1+x)^a &= 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + ... \\[2em]
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n &|x|\lt1 \\[1ex]
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<ol>
<li>无穷级数求和函数，应灵活运用上面的泰勒级数展开式</li>
<li>在与导数相关题目中的应用：</li>
</ol>
<p>【2016年考研数一16题】设函数 <script type="math/tex">f(x)=\arctan x-\cfrac{x}{1+ax^2}</script>，且 <script type="math/tex">f'''(0)=1</script>，则 <script type="math/tex">a=</script> ____<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
f(x)&=\arctan x-\cfrac{x}{1+ax^2} \\
&=x-\cfrac{1}{3}x^3+o(x^3)-x(1-ax^2+o(x^2)) \\
&=(a-\cfrac{1}{3})x^3+o(x^3) \\[1ex]
f'''(x)&=3*2*1*(a-\cfrac{1}{3})=1 \\[1ex]
a&=\cfrac{1}{2}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h3 id="3_2">3. 无穷比介小</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th></th>
<th>表达式</th>
<th>
<script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  的</th>
<th>趋向于0的速</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 0 </script>
</td>
<td>高阶无穷小</td>
<td>
<script type="math/tex"> \alpha(x) 快于 \beta(x) </script>
</td>
<td>
<script type="math/tex"> \alpha(x) = o(\beta(x)) </script>
</td>
</tr>
<tr>
<td>2</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = \infty </script>
</td>
<td>低阶无穷小</td>
<td>
<script type="math/tex"> \alpha(x) 慢于 \beta(x) </script>
</td>
<td>
<script type="math/tex"> \alpha(x) = \omega(\beta(x)) </script>
</td>
</tr>
<tr>
<td>3</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = c \neq 0 </script>
</td>
<td>同阶无穷小</td>
<td>相近</td>
<td></td>
</tr>
<tr>
<td>4</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 1 </script>
</td>
<td>等阶无穷小</td>
<td>相等</td>
<td>
<script type="math/tex"> \alpha(x) ～ \beta(x) </script>
</td>
</tr>
<tr>
<td>5</td>
<td>
<script type="math/tex"> \lim\cfrac{\alpha(x)}{[\beta(x)]^k} = c \neq 0，k \gt 0 </script>
</td>
<td>
<script type="math/tex"> k </script>  阶无穷小</td>
<td></td>
<td></td>
</tr>
</tbody>
</table></div>
<h3 id="4_2">4.  无穷小的运算规则</h3>
<ol>
<li>有限个无穷小的和是无穷小</li>
<li>有限个无穷小的乘积是无穷小</li>
<li>有界函数与无穷小的乘积是无穷小</li>
<li>无穷小的运算</li>
<li>加减法：低价吸收高阶  <script type="math/tex"> o(x^2) \pm o(x^3) = o(x^2) </script>
</li>
<li>乘法：阶数累加  <script type="math/tex"> o(x^2) * o(x^3) = o(x^5) </script>
</li>
<li>非0常数相乘不影响阶数   <script type="math/tex"> o(k * x^2) = k * o(x^2) </script>
</li>
</ol>
<h3 id="5_1">5. 常用的等价无穷小</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
   \sin x &\sim x \\[1em]
   \tan x &\sim x \\[1em]
\arcsin x &\sim x \\[1em]
\arctan x &\sim x \\[2em]
\ln(1 + x) &\sim x \\
\log_a(1 + x) &\sim \cfrac{x}{\ln a} \\[1em]
  e^x - 1 &\sim x \\[1ex]
  a^x - 1 &\sim x\ln a \\[2em]
(1 + x)^a - 1&\sim ax \\[1em]
1 -\cos\ x &\sim \frac{1}{2}x^2
\end{split}\end{equation}
</script>
</p>
<h3 id="6">6. 函数的连续与间断</h3>
<ol>
<li>可去间断点</li>
<li>跳跃间断点</li>
<li>无穷间断点</li>
<li>震荡间断点</li>
</ol>
<h2 id="3-4">第3-4讲 一元函数微分学</h2>
<h3 id="1_4">1. 导数定义</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f'(x_0) &= \lim_{\Delta x \to 0}\frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \\[3ex]
&= \lim_{x \to x_0}\frac{f(x) - f(x_0)}{x - x_0} \\
\end{split}\end{equation}
</script>
</p>
<p>无穷导数  <script type="math/tex"> \infty </script>  视为倒数不存在</p>
<blockquote class="content-quote">
<p>常用性质：<script type="math/tex"> f(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f'(x) </script> 为奇函数 <script type="math/tex"> \Rightarrow f''(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f^{(3)}(x) </script> 为奇函数&hellip;</p>
<p>极限存在 ⟺ <script type="math/tex">f_-'(x_0)=f_+'(x_0)</script>
</p>
</blockquote>
<h3 id="2_3">2. 微分定义</h3>
<p>设函数 <script type="math/tex">y=f(x)</script> 在点 <script type="math/tex">x_0</script> 的某领域内有定义，且 <script type="math/tex">x_0+\Delta x</script> 在该领域内，对于函数增量<br />
<script type="math/tex; mode=display">
\Delta y=f(x_0+\Delta x)-f(x_0)
</script>
<br />
若存在与 <script type="math/tex">\Delta x</script>
<strong>无关</strong>的，而仅与 <script type="math/tex">x</script>
<strong>有关</strong> 的常数 <script type="math/tex">A</script>，使得<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\Delta y &= A\Delta x+o(\Delta x) \\[1ex]
&= 线性增量 + 高阶无穷小量
\end{split}\end{equation}
</script>
<br />
则称 <script type="math/tex">f(x)</script> 在 <script type="math/tex">x_0</script> 处 <strong>可微</strong>，并称 <script type="math/tex">A\Delta x</script> 为 <script type="math/tex">f(x)</script> 在点 <script type="math/tex">x_0</script> 处的 <strong>微分</strong>，记作<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dy\bigg|_{x=x_0} &= A\Delta x \\[1ex]
&= Adx \\[1em]
导数：f'(x_0) = \cfrac{dy}{dx}\bigg|_{x=x_0} &= A
\end{split}\end{equation}
</script>
</p>
<h3 id="3_3">3. 导数与微分的计算</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
积的导数： [u(x)v(x)]' &= u'(x)v(x) + u(x)v'(x) \\[2ex]
积的微分： d[u(x)v(x)] &= du(x)v(x) + u(x)dv(x) \\[4ex]
商的导数： \bigg[\cfrac{u(x)}{v(x)}\bigg]' &= \cfrac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}, v(x) \neq 0 \\[2ex]
商的微分： d\bigg[\cfrac{u(x)}{v(x)}\bigg] &= \cfrac{du(x)v(x) - u(x)dv(x)}{[v(x)]^2}, v(x) \neq 0 \\[4ex]
复合函数的导数： \{f[g(x)]\}' &= f'[g(x)]g'(x) \\[2ex]
复合函数的微分： d\{f[g(x)]\} &= f'[g(x)]g'(x)dx \\[4ex]
\end{split}\end{equation}
</script>
</p>
<h3 id="4_3">4. 反函数求导</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y_x'&=\cfrac{1}{x_y'} \\[2ex]
推导过程：x_y'&=\cfrac{dx}{dy}=\cfrac{1}{\cfrac{dy}{dx}}=\cfrac{1}{y_x'} \\[1ex]
x_{yy}''&=\cfrac{d\cfrac{dx}{dy}}{dy}=\cfrac{d\cfrac{1}{y_x'}}{dy}=\cfrac{d\cfrac{1}{y_x'}}{dx}·\cfrac{1}{y_x'} \\[1ex]
&= -\cfrac{y_{xx}''}{(y_x')^2}·\cfrac{1}{y_x'} \\[1ex]
&=-\cfrac{y_{xx}''}{(y_x')^3}\\[2em]
y_x'&=\cfrac{1}{x_y'} \\
y_{xx}''&=-\ x_{yy}''·(y_x')^3=-\ \cfrac{x_{yy}''}{(x_y')^3}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>示例：求反函数 <script type="math/tex">y=\cfrac{1}{a}\arctan\cfrac{1}{a}x</script> 的导数，易得原函数 <script type="math/tex">y=a\tan ax</script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{1}{a}(\arctan\cfrac{1}{a}x)'
&=\cfrac{1}{(a\tan ay)'} \\[1ex]
&=\cfrac{1}{a(\cfrac{\sin ay}{\cos ay})'} \\[1ex]
&=\cfrac{1}{a(\cfrac{a\cos ay\cos ay+a\sin ay\sin ay}{\cos^2 ay})} \\[1ex]
&=\cfrac{\cos^2 ay}{a^2} \\[1ex]
&=\cfrac{\cos^2 ay}{a^2(\cos^2 ay + \sin^2 ay)} \\[1ex]
&=\cfrac{1}{a^2(1 + \tan^2 ay)} \\[1ex]
&=\cfrac{1}{a^2(1 + \cfrac{1}{a^2}x^2)} \\[1ex]
&=\cfrac{1}{a^2 + x^2} \\[1ex]
另外,\ \ \ \ \lim_{x\to0}\cfrac{\cfrac{\cos^2 ax}{a^2}}{\cfrac{1}{a^2+x^2}}&=(1+\cfrac{1}{a^2}x^2)\cos^2 ax=1 \\[1ex]
可见,\ \ \ \ \cfrac{cos^2 x}{a^2}\ 与&\ \cfrac{1}{a^2+x^2}\ 为等价无穷小
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h3 id="5_2">5. 幂指函数求导</h3>
<p>
<script type="math/tex; mode=display">
u(x)^{v(x)}=e^{v(x)\ln u(x)} \\[2em]
\{u(x)\gt0, u(x)\neq1\}
</script>
</p>
<h3 id="6_1">6. 参数方程求导</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dy}{dx} &= \cfrac{\cfrac{dy}{dt}}{\cfrac{dx}{dt}} = \cfrac{\psi'(t)}{\varphi'(t)} \\[2ex]
\cfrac{d^2y}{dx^2} &= \cfrac{d(\cfrac{dy}{dx})}{dx} = \cfrac{\cfrac{d(\cfrac{dy}{dx})}{dt}}{\cfrac{dx}{dt}}= \cfrac{\psi''(t)\varphi'(t)-\psi'(t)\varphi''(t)}{[\varphi'(t)]^3}
\end{split}\end{equation}
</script>
</p>
<h3 id="7">7. 莱布尼茨公式</h3>
<p>
<script type="math/tex; mode=display">
(uv)^{(n)} = u^{(n)}v + C^1_nu^{(n-1)}v' + \cdots + C^{n-1}_nu'v^{(n-1)} + uv^{(n)}
</script>
</p>
<h3 id="8">8. 可微、可导、连续、可积的关系</h3>
<ul>
<li>
<p>可微 ⟺ 可导 ⟹ 连续 ⟹ 可积</p>
</li>
<li>
<p>可导的条件：（左极限要等于右极限）</p>
<ul>
<li>如 <script type="math/tex"> y = |x|, \lim\limits_{x \to 0^-} = -1, \lim\limits_{x \to 0^+} = 1, 在\ x = 0\ 处不可导 </script>
</li>
</ul>
</li>
</ul>
<h2 id="5-7">第5-7讲 一元函数微分学的应用</h2>
<h3 id="1_5">1. 极值、单调性</h3>
<h4 id="1-yx">（1）一元函数 <script type="math/tex">y(x)</script> 的极值</h4>
<h4 id="2-fxy0-yx-yx">（2）一元函数隐函数 <script type="math/tex"> F(x,y)=0 </script> 确定 <script type="math/tex">y=(x)</script>，求 <script type="math/tex">y(x)</script> 的极值</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F_x'+F_y'y_x'&=0 \\[1ex]
\Rightarrow
y_x'&=-\cfrac{F_x'}{F_y'}=0 \\[1ex]
\Rightarrow
F_x'&=0\ \ \ \ 得到极值点 \\[2em]
y_x''&=-\cfrac{F_{xx}''F_y'-F_x'F_{yy}''y_x'}{(F_y')^2} \\[1ex]
&=-\cfrac{F_{xx}''}{F_y'}\ \ \ \ 判断极大值极小值
\end{split}\end{equation}
</script>
</p>
<h3 id="2_4">2. 拐点、凹凸性</h3>
<ul>
<li>拐点：凹凸性改变的分界点</li>
<li>拐点存在的 <strong>必要条件</strong>：<script type="math/tex">f''(x_0)=0</script>
</li>
</ul>
<h3 id="3_4">3. 渐近线</h3>
<h4 id="1_6">（1）水平渐近线和铅直渐近线</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}[f(x)] &= y_0,\ 则y=y_0为水平渐近线 \\[2ex]
实际上是求 \lim_{x\to\infty}[f(x)-y_0] &= 0 \\[1em]
\lim_{x\to x_0}[f(x)]&=\infty,\ 则x=x_0为铅直渐近线 
\end{split}\end{equation}
</script>
</p>
<h4 id="2x">（2）斜渐近线的正确求法(在x趋向于无穷时)</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}(\cfrac{f(x)}{x}) &= A \\[1ex]
\lim_{x\to\infty}[f(x)-Ax] &= B \\[1ex]
渐近线方程为\ y &= Ax + B \\[1em]
实际上是求\ \lim_{x\to\infty}[f(x)-(Ax+B)] &= 0
\end{split}\end{equation}
</script>
</p>
<h3 id="4_4">4. 曲率与曲率半径</h3>
<p><img class="pure-img" alt="qulv" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/qulv.png" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
平均曲率：\overline k &= \bigg|\cfrac{\Delta \alpha}{\Delta s}\bigg| \\[1ex]
曲率：k &= \lim_{\Delta s \to 0}\bigg|\cfrac{\Delta \alpha}{\Delta s}\bigg| \\[1ex]
y' &= \tan\alpha \\[1ex]
y'' = \sec^2\alpha \cfrac{d\alpha}{dx} ⟺ d\alpha &= \cfrac{y''}{1+\tan^2\alpha}dx \\[1ex]
&= \cfrac{y''}{1+(y')^2}dx\\[1ex]
ds &= \sqrt{1+(y')}dx \\[1ex]
曲率：k&=\cfrac{|y''|}{[1+(y')^2]^{\frac{3}{2}}} \\[1em]
曲率半径：R&=\cfrac{1}{k}
\end{split}\end{equation}
</script>
</p>
<h3 id="5_3">5. 中值定理</h3>
<h4 id="1_7">定理1（费马定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
可导 \\
取极值
\end{cases}
，则f'(x)=0
</script>
</p>
<h4 id="2_5">定理2（罗尔定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
f(a) = f(b)
\end{cases}
，则存在\xi\in(a,b)，使得f'(\xi)=0
</script>
</p>
<h4 id="3_5">定理3（拉格朗日中值定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导
\end{cases}
，则存在\xi\in(a,b)，使得 \\
f(b) - f(a) = f'(\xi)(b - a) \\ 
即f'(\xi) = \cfrac{f(b) - f(a)}{b - a}
</script>
</p>
<h4 id="4_5">定理4（柯西中值定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x),g(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
g'(x)\ne0
\end{cases}
，则存在\xi\in(a,b)，使得 \\ 
\cfrac{f'(\xi)}{g'(\xi)} = \cfrac{f(b) - f(a)}{g(b) - g(a)}
</script>
</p>
<h4 id="5_4">定理5（泰勒中值定理）（泰勒公式 / 麦克劳林公式）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n \\[2em]
(带拉格朗日余项)\ \ f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - x_0)^{n+1} \\[2ex]
(皮亚诺余项)\ \ f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + o((x - x_0)^{n})
\end{split}\end{equation}
</script>
</p>
<h4 id="6_2">定理6（积分中值定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x)在[a,b]上连续，则存在\xi\in[a,b]，使得 \\ 
\int_{a}^{b}f(x)dx = f(\xi)(b - a)
</script>
</p>
<h2 id="8-9">第8-9讲 一元函数积分学</h2>
<h3 id="1_8">1. 公式</h3>
<h4 id="1_9">（1）重要的公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
                                 \int a^xdx &= \cfrac{a^x}{\ln a} + C \\[2em]
        (*\ x=a\tan t)\ \ \ \ \int\cfrac{1}{x^2+a^2}dx &= \cfrac{1}{a}\arctan\cfrac{x}{a} + C               & (a\gt0) \\[1ex]
        (*)\ \ \ \ \int\cfrac{1}{x^2-a^2}dx &= \cfrac{1}{2a}\ln\bigg|\cfrac{x-a}{x+a}\bigg| + C  & (a\gt0) \\[1ex]
       (*)\ \ \ \ \int\cfrac{1}{-x^2+a^2}dx &= \cfrac{1}{2a}\ln\bigg|\cfrac{x+a}{-x+a}\bigg| + C & (a\gt0) \\[2em]
 (*)\ \ \ \ \int\cfrac{1}{\sqrt{x^2+a^2}}dx &= \ln\bigg|x+\sqrt{x^2+a^2}\bigg| + C \\[1ex]
 (*)\ \ \ \ \int\cfrac{1}{\sqrt{x^2-a^2}}dx &= \ln\bigg|x+\sqrt{x^2-a^2}\bigg| + C &\ \ \ \ (|x|\gt|a|) \\[1ex]
(*\ x=a\sin t)\ \ \ \ \int\cfrac{1}{\sqrt{-x^2+a^2}}dx &= \arcsin\cfrac{x}{a} + C             & (a\gt0) \\[1ex]
                                            &=-\arccos\cfrac{x}{a} + C             & (a\gt0)
\end{split}\end{equation}
</script>
</p>
<h4 id="2_6">（2）被积函数包含三角函数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
     \int\csc xdx = \int\cfrac{1}{\sin x}dx &=-\ln\bigg|\csc x + \cot x\bigg| + C \\[1ex]
     \int\sec xdx = \int\cfrac{1}{\cos x}dx &= \ln\bigg|\sec x + \tan x\bigg| + C \\[2em]
                              \int\tan  xdx &=-\ln\bigg|\cos x\bigg| + C \\[1ex]
     \int\cot xdx = \int\cfrac{1}{\tan x}dx &= \ln\bigg|\sin x\bigg| + C \\[2em]
                              \int\sin^2xdx &= \cfrac{x}{2} - \cfrac{\sin 2x}{4} + C \\[1ex]
                              \int\cos^2xdx &= \cfrac{x}{2} + \cfrac{\sin 2x}{4} + C \\[1ex]
     \int\csc^2x = \int\cfrac{1}{\sin^2}xdx &=-\cot x + C \\[1ex]
     \int\sec^2x = \int\cfrac{1}{\cos^2}xdx &= \tan x + C \\[2em]
                              \int\tan^2xdx &= \tan x - x + C \\[1ex]
     \int\cot^2x = \int\cfrac{1}{\tan^2x}dx &=-\cot x - x + C \\[2em]
\int\csc x\cot xdx = \int\cfrac{\cos x}{\sin^2x}dx &=-\csc x + C \\[1ex]
\int\sec x\tan xdx = \int\cfrac{\sin x}{\cos^2x}dx &= \sec x + C \\[2em]
\end{split}\end{equation}
</script>
</p>
<h3 id="2_7">2. 不定积分的计算方法</h3>
<h4 id="1_10">（1）凑微</h4>
<h4 id="2_8">（2）换元</h4>
<h4 id="3_6">（3）分部积分</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int u(x)v'(x)dx &= \int u(x)dv(x) \\
&= u(x)v(x) - \int u'(x)v(x)dx
\end{split}\end{equation}
</script>
</p>
<h3 id="3_7">3. 定积分的计算</h3>
<h4 id="1_11">（1）定积分的定义</h4>
<h5 id="1_12">1、均匀分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{\cfrac{1}{n},\ \cfrac{1}{n},\ \cfrac{1}{n}\cdots,\ \cfrac{1}{n}\bigg\} \\[2ex]
[x_{k-1},x_{k}]&=[\cfrac{k-1}{n},\cfrac{k}{n}] \\[2ex]
每段\ x_{k}-x_{k-1}&=\cfrac{1}{n} \\[2ex]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的端点 \\[2ex]
(1)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{k}{n})\cfrac{1}{n}} \\[2ex]
(2)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{k-1}{n})\cfrac{1}{n}} \\[3em]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的中点 \\[1ex]
&\xi_k=\cfrac{x_{k-1}+x_{k}}{2}=\cfrac{2k-1}{2n} \\[2ex]
(3)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{2k-1}{2n})\cfrac{1}{n}} \\[3em]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的几何平均 \\[1ex]
&\xi_k=\sqrt{x_{k-1}x_{k}}=\cfrac{\sqrt{(k-1)k}}{n} \\[2ex]
(4)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{\sqrt{(k-1)k}}{n})\cfrac{1}{n}}
\end{split}\end{equation}
</script>
</p>
<h5 id="2_9">2、等差分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{1l,\ 2l,\ 3l,\ \cdots,\ (k-1)l,\ kl,\ \cdots,\ nl\bigg\} \\[2ex]
[x_{k-1},x_{k}]
=&[\cfrac{\cfrac{(k-1)k}{2}l}{\cfrac{n(1+n)}{2}l},\cfrac{\cfrac{k(1+k)}{2}l}{\cfrac{n(1+n)}{2}l}] \\[2ex]
&=[\cfrac{(k-1)k}{n(1+n)},\cfrac{k(1+k)}{n(1+n)}] \\[3ex]
每段\ x_{k}-x_{k-1}&=\cfrac{2k}{n(1+n)}
\end{split}\end{equation}
</script>
</p>
<h5 id="3_8">3、等比分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{2^0l,\ 2^1l,\ 2^2l,\ \cdots,\ 2^{k-2}l,\ 2^{k-1}l, \ \cdots\ 2^{n-1}l\bigg\} \\[2ex]
[x_{k-1},x_{k}]
&=[\cfrac{\cfrac{2^0(1-2^{k-1})}{1-2}l}{\cfrac{2^0(1-2^n)}{1-2}l},
\cfrac{\cfrac{2^0(1-2^k)}{1-2}l}{\cfrac{2^0(1-2^n)}{1-2}l}] \\[2ex]
&=[\cfrac{2^{k-1}-1}{2^n-1},
\cfrac{2^k-1}{2^n-1}] \\[3ex]
每段\ x_{k}-x_{k-1}
&=\cfrac{2^k-2^{k-1}}{2^n-1} \\[2ex]
&=\cfrac{2^{k-1}}{2^n-1}
\end{split}\end{equation}
</script>
</p>
<h4 id="2_10">（2）重要公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(区间再现公式)\int_a^b f(x)dx &= \int_a^b f(a+b-x)dx \\[1ex]
                &= \cfrac{1}{2}\int_a^b \bigg[f(x) + f(a+b-x)\bigg]dx \\[1ex]
                &= \int_a^{\frac{a+b}{2}} \bigg[f(x) + f(a+b-x)\bigg]dx \\[2em]
(区间简化公式)\ \ \ x-\cfrac{a+b}{2}&=\cfrac{b-a}{2}\sin t \\[1ex]
\int_a^b f(x)dx &= \int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\bigg[f(\cfrac{a+b}{2}+\cfrac{b-a}{2}\sin t)\cfrac{b-a}{2}\cos t\bigg]dt \\[2em]
(区间简化公式)\ \ \ x-a&=(b-a)t \\[1ex]
\int_a^b f(x)dx &= \int_0^1 (b-a)f[a+(b-a)t]dt \\[2em]
\int_{-a}^af(x)dx &= \int_0^a[f(x)+f(-x)]dx \ \ \ \ \ \ (a>0)
\end{split}\end{equation}
</script>
</p>
<h4 id="3_9">（3）华式公式（“点火公式”）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_0^{\frac{\pi}{2}}\sin^nx\ dx &= \\[1ex]
\int_0^{\frac{\pi}{2}}\cos^nx\ dx &= 
\begin{cases}
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{2}{3} &n为大于1的奇数  \\[2ex]
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{1}{2}·\cfrac{\pi}{2} &n为正偶数
\end{cases}  \\[2em]
\int_0^{\pi}\sin^nx\ dx &= 
\begin{cases}
2\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正奇数  \\[1ex]
2\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正偶数
\end{cases} \\[1ex]
\int_0^{\pi}\cos^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
2\int_0^{\frac{\pi}{2}}\cos^nx\ dx &n为正偶数
\end{cases}  \\[2em]
\int_0^{2\pi}\sin^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
4\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正偶数
\end{cases}  \\[1ex]
\int_0^{2\pi}\cos^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
4\int_0^{\frac{\pi}{2}}\cos^nx\ dx &n为正偶数
\end{cases}  \\[2em]
\int_0^{\pi}xf(\sin x)dx &= \cfrac{\pi}{2}\int_0^{\pi}f(\sin x)dx = \pi\int_0^{\frac{\pi}{2}}xf(\sin x)dx \\[1ex]
\int_0^{nT}xf(x)dx &= \cfrac{n^2 T}{2}\int_0^T f(x)dx\\[2em]            
\int_0^{\frac{\pi}{2}}f(\sin x)dx &= \int_0^{\frac{\pi}{2}}f(\cos x)dx \\[1ex]
\int_0^{\frac{\pi}{2}}f(\sin x, \cos x)dx &= \int_0^{\frac{\pi}{2}}f(\cos x, \sin x)dx \\[2em]
\end{split}\end{equation}
</script>
</p>
<h4 id="4_6">（4）伽马函数</h4>
<p>
<script type="math/tex; mode=display">
实数域：\Gamma(x) = \int_0^{+\infty}t^{x-1}e^{-t}dt,\ (x>0)
</script>
</p>
<p>伽马函数的推导<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n (对比)\\[1ex]
&= \int_0^{+\infty}e^{-(1-x)t}dt \\[1ex]
&= \int_0^{+\infty}e^{-t+xt}dt \\[1ex]
&= \int_0^{+\infty}e^{-t}\sum_{n=0}^\infty \cfrac{(xt)^n}{n!}dt \\[1ex]
&= \sum_{n=0}^\infty \cfrac{\int_0^{+\infty}t^ne^{-t} dt}{n!}x^n (对比)\\[2em]
\int_0^{+\infty}t^ne^{-t} dt &= n!\ \ \ \ 【便利公式】
\end{split}\end{equation}
</script>
</p>
<h3 id="4_7">4. 变限积分的计算</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\bigg[\int_a^{\varphi(x)}f(t)dt\bigg]_x' &= f[\varphi(x)]·\varphi'(x) \\[1ex]
\bigg[\int_{\varphi_1(x)}^{\varphi_2(x)}f(t)dt\bigg]_x' &= f[\varphi_2(x)]·\varphi_2'(x) - f[\varphi_1(x)]·\varphi_1'(x)
\end{split}\end{equation}
</script>
</p>
<h3 id="5_5">5. 反常积分的计算</h3>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/1203675-20171207172202738-17246288.png" alt="1203675-20171207172202738-17246288" style="zoom:67%;" /></p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/1203675-20171207172925972-145324329.png" alt="1203675-20171207172925972-145324329" style="zoom:67%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_a^{+\infty}f(x)dx &= \lim_{x\to+\infty}F(x)-F(a) \\[1em]
(a为瑕点)\int_a^b f(x)dx &= F(b)-\lim_{x\to a}F(x)
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases} \\[2em]
广义\ p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^\alpha(\ln x)^\beta}dx
&\begin{cases}
\text{收敛}, &\alpha\gt1\ 或\ \alpha=1,\beta\gt1 \\[2ex]
\text{发散}, &\alpha\lt1\ 或\ \alpha=1,\beta\le1 \\[2ex]
\end{cases} \\[2em]
瑕积分：
\int_0^1\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & 【\ q\lt1\ 】 \\[2ex]
\text{发散}, & 【\ q\ge1\ 】
\end{cases}
\end{split}\end{equation}
</script>
</p>
<h2 id="10-12">第10-12讲 一元函数积分学的应用</h2>
<p>一型曲线积分&hellip;</p>
<h2 id="13">第13讲 多元函数微分学</h2>
<h3 id="1_13">1. 导数与微分</h3>
<p>
<script type="math/tex; mode=display">
偏导数存在 \Rightarrow 可微
\begin{cases}
\Rightarrow 偏导数存在 &（某方向双侧）\\[2ex]
\Rightarrow 连续 \Rightarrow 极限存在 &（全方向） \\[2ex]
\Rightarrow 方向导数存在 &（某方向单侧）
\end{cases}
</script>
</p>
<h4 id="1_14">（1）偏导数的定义公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f'_x(x,y)
&= \lim_{\Delta x\to0}\cfrac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x} \\[3ex]
&= \lim_{x \to x_0}\frac{f(x,y_0) - f(x_0,y_0)}{x - x_0} \\
\end{split}\end{equation}
</script>
</p>
<h4 id="2_11">（2）二元函数微分的定义</h4>
<p>设函数 <script type="math/tex">z=f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 的某领域内有定义，且 <script type="math/tex">(x_0+\Delta x,y_0+\Delta y)</script> 在该领域内，对于 <strong>全增量</strong><br />
<script type="math/tex; mode=display">
\Delta z = f(x_0+\Delta x, y_0+\Delta y)-f(x_0,y_0)
</script>
<br />
若存在与 <script type="math/tex">\Delta x, \Delta y</script>
<strong>无关</strong>，而仅与 <script type="math/tex">x,y</script>
<strong>有关</strong> 的常数 <script type="math/tex">A,B</script> 使得<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\Delta z &= A\Delta x + B\Delta y+o(\sqrt{(\Delta x)^2+(\Delta y)^2}) \\[1ex]
&= 线性增量 + 高阶无穷小量
\end{split}\end{equation}
</script>
<br />
则称 <script type="math/tex">f(x,y)</script> 在 <script type="math/tex">(x_0,y_0)</script> 处 <strong>可微</strong>，并称 <script type="math/tex">A\Delta x + B\Delta y</script> 为 <script type="math/tex">f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 处的 <strong>全微分</strong>，记作<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dz\bigg|_{(x,y)=(x_0,y_0)} &= A\Delta x + B\Delta y\\[1ex]
&= Adx + Bdy\\[1em]
偏导数：f_x'(x_0,y_0) = \cfrac{dz}{dx}\bigg|_{(x,y)=(x_0,y_0)} &= A \\[1ex]
偏导数：f_y'(x_0,y_0) = \cfrac{dz}{dy}\bigg|_{(x,y)=(x_0,y_0)} &= B
\end{split}\end{equation}
</script>
<br />
<script type="math/tex">z=f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 处可微的条件<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
（可微的条件）
 &\lim_{\Delta x \to 0,\Delta y \to 0}\cfrac{\Delta z - (A\Delta x + B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}} \\[1ex]
=&\lim_{\Delta x \to 0,\Delta y \to 0}\cfrac{\Delta z - [f'_x(x_0,y_0)\Delta x + f'_y(x_0,y_0)\Delta y]}{\sqrt{(\Delta x)^2+(\Delta y)^2}} \\[1ex]
=&0
\end{split}\end{equation}
</script>
</p>
<p>偏导数连续的条件（<strong>注意是从各个方向靠近</strong>）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{(x,y)\to(x_0,y_0)}f_x'(x,y)=f_x'(x_0,y_0) \\[1ex]
\lim_{(x,y)\to(x_0,y_0)}f_y'(x,y)=f_y'(x_0,y_0)
\end{split}\end{equation}
</script>
</p>
<h3 id="2_12">2. 复合函数求导法</h3>
<h4 id="1_15">（1）链式求导法则</h4>
<p>设 <script type="math/tex"> z=z(u,v),u=u(x,y),v=v(x,y) </script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{pmatrix}
\cfrac{\partial z}{\partial x} &\cfrac{\partial z}{\partial y} \\
\end{pmatrix} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} &\cfrac{\partial u}{\partial y} \\
\cfrac{\partial v}{\partial x} &\cfrac{\partial v}{\partial y} \\
\end{pmatrix} \\[1ex]
\cfrac{\partial z}{\partial x} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x} \\[1ex]
\cfrac{\partial z}{\partial y} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial y}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial y}
\end{split}\end{equation}
</script>
</p>
<h4 id="2_13">（2）全导数</h4>
<p>设 <script type="math/tex"> z=z(u,v),u=u(x),v=v(x) </script> ，即 <script type="math/tex"> z </script> 最终是 <script type="math/tex"> x </script> 的函数，则 <script type="math/tex"> \cfrac{dz}{dx} </script> 叫 <strong>全导数</strong><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dz}{dx} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} \\ 
\cfrac{\partial v}{\partial x}
\end{pmatrix} \\[1ex]
&= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x}
\end{split}\end{equation}
</script>
</p>
<h4 id="3_10">（3）全微分形式不变性</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dz &= \cfrac{\partial z}{\partial x}dx + \cfrac{\partial z}{\partial y}dy\\
   &= \cfrac{\partial z}{\partial u}du + \cfrac{\partial z}{\partial v}dv
\end{split}\end{equation}
</script>
</p>
<h3 id="3_11">3. 隐函数求导法</h3>
<h4 id="1_16">（1）一个方程的情形</h4>
<p>将 <script type="math/tex">z=z(x,y)</script> 转化为 <script type="math/tex">F(x,y,z)=0</script> 求偏导<br />
<script type="math/tex; mode=display">
\cfrac{\partial z}{\partial x} = -\cfrac{F'_x}{F'_z}\\
\cfrac{\partial z}{\partial y} = -\cfrac{F'_y}{F'_z}
</script>
</p>
<h4 id="2_14">（2）方程组情形</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{cases}
F(x,u,v)=0 \\[1ex]
G(x,u,v)=0
\end{cases}
，当满足
\left|\begin{array}{c c}
\cfrac{\partial F}{\partial u} & \cfrac{\partial F}{\partial v} \\
\cfrac{\partial G}{\partial u} & \cfrac{\partial G}{\partial v} \\
\end{array}\right|
=\cfrac{\partial(F,G)}{\partial(u,v)}\neq0时，可以确定
\begin{cases}
u=u(x) \\[1ex]
v=v(x)
\end{cases}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\cfrac{du}{dx}=-\cfrac{\cfrac{\partial(F,G)}{\partial(u,v)}}{\cfrac{\partial(F,G)}{\partial(u,v)}} 
\ \ \ \ \ \ \ \ \ 
\cfrac{dv}{dx}=-\cfrac{\cfrac{\partial(F,G)}{\partial(u,v)}}{\cfrac{\partial(F,G)}{\partial(u,v)}}
</script>
</p>
<blockquote class="content-quote">
<p>推导过程：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
\cfrac{\partial F}{\partial x} + \cfrac{\partial F}{\partial u}\cfrac{du}{dx} + \cfrac{\partial F}{\partial v}\cfrac{dv}{dx} = 0 \\[1ex]
\cfrac{\partial G}{\partial x} + \cfrac{\partial G}{\partial u}\cfrac{du}{dx} + \cfrac{\partial G}{\partial v}\cfrac{dv}{dx} = 0 \\[1ex]
\end{cases} \\[2ex]
⟹
&\begin{cases}
\cfrac{\partial F}{\partial u}\cfrac{du}{dx} + \cfrac{\partial F}{\partial v}\cfrac{dv}{dx} = -\cfrac{\partial F}{\partial x} \\[1ex]
\cfrac{\partial G}{\partial u}\cfrac{du}{dx} + \cfrac{\partial G}{\partial v}\cfrac{dv}{dx} = -\cfrac{\partial G}{\partial x} \\[1ex]
\end{cases} \\[2ex]
⟹
&\begin{cases}
a_{11}\cfrac{du}{dx} + a_{12}\cfrac{dv}{dx} = b_1 \\[1ex]
a_{21}\cfrac{du}{dx} + a_{22}\cfrac{dv}{dx} = b_2 \\[1ex]
\end{cases}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
(克拉默法则)
\ \ \ \ \ \ \ \ \ 
\cfrac{du}{dx}=
-\cfrac
{\left|\begin{array}{c c}
b_1 & a_{12} \\
b_2 & a_{22} \\
\end{array}\right|}
{\left|\begin{array}{c c}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array}\right|}
\ \ \ \ \ \ \ \ \ 
\cfrac{dv}{dx}=
-\cfrac
{\left|\begin{array}{c c}
a_{11} & b_1 \\
a_{21} & b_2 \\
\end{array}\right|}
{\left|\begin{array}{c c}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array}\right|}
</script>
</p>
</blockquote>
<h3 id="4-fxy">4. 二元函数 <script type="math/tex">f(x,y)</script> 极值</h3>
<p>先求一阶偏导数 <br />
<script type="math/tex; mode=display">
f'_x(x,y) \\[1ex]
f'_y(x,y)
</script>
</p>
<p>再求二阶偏导数</p>
<p>
<script type="math/tex; mode=display">
A = f''_{xx}(x_0, y_0) \\[1ex]
B = f''_{xy}(x_0, y_0) \\[1ex]
C = f''_{yy}(x_0, y_0)
</script>
</p>
<p>
<script type="math/tex"> (x_0, y_0) </script>  为一阶导数  <script type="math/tex"> f_x(x,y) = 0，f_y(x,y) = 0 </script>  的点</p>
<ol>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \gt 0 </script>  极小值点</li>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \lt 0 </script>  极大值点</li>
<li>
<script type="math/tex"> AC-B^2 \lt 0 </script>  非极值点（鞍点）</li>
<li>
<script type="math/tex"> AC-B^2 = 0 </script>  不确定</li>
</ol>
<h3 id="5-fxyz0-zxy-zxy">5. 多元函数隐函数 <script type="math/tex"> F(x,y,z)=0 </script> 确定 <script type="math/tex">z(x,y)</script>，求 <script type="math/tex">z(x,y)</script> 的极值</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{cases}
F_x'+F_z'z_x'=0 \\[6ex]
F_y'+F_z'z_y'=0
\end{cases}
\Rightarrow
\begin{cases}
z_x' = -\cfrac{F_x'}{F_z'} = 0 \\[2ex]
z_y' = -\cfrac{F_y'}{F_z'} = 0
\end{cases}
&\Rightarrow \begin{cases}
F_x' = 0 \\[6ex]
F_y' = 0
\end{cases} \\[4ex]
&\Rightarrow P_0(x_0,y_0)、P_1(x_1,y_1)，...\ \ \ \ 得到极值点\\[2em]
A = z_{xx}'' 
&= \cfrac{d(z_x')}{dx} \\[1ex]
&= \cfrac{d(-\cfrac{F_x'}{F_z'})}{dx} \\[1ex]
&= -\cfrac{(F_{xx}''+F_{xz}''z_x')F_z'-F_x'(F_{zx}''+F_{zz}''z_x')}{(F_z')^2} \\[1ex]
&= -\cfrac{F_{xx}''}{F_z'} \\[2ex]
B = z_{xy}''
&= -\cfrac{F_{xy}''}{F_z'} \\[2ex]
C = z_{xy}''
&= -\cfrac{F_{yy}''}{F_z'} \ \ \ \ 判断极大值极小值
\end{split}\end{equation}
</script>
</p>
<h3 id="6_3">6. 条件极值（拉格朗日数乘法）</h3>
<h4 id="fxy">【二元】函数 <script type="math/tex">f(x,y)</script> 在条件</h4>
<p>
<script type="math/tex; mode=display">
\varphi(x,y) =0
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
F(x,y,\lambda) = f(x,y)+\lambda\varphi(x,y) \\[2em]
\begin{cases}
F'_x(x,y,\lambda) = f'_x(x,y) + \lambda \varphi'_x(x,y) = 0 \\[2ex]
F'_y(x,y,\lambda) = f'_y(x,y) + \lambda \varphi'_y(x,y) = 0 \\[2ex]
F'_\lambda(x,y,\lambda) = \varphi(x,y)= 0 \\
\end{cases}
</script>
</p>
<h4 id="fxyz">【三元】函数 <script type="math/tex">f(x,y,z)</script> 在条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{cases}
\varphi(x,y,z)=0 \\[2ex]
\psi(x,y,z)=0
\end{cases}
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
F(x,y,z,\lambda,\mu)=f(x,y,z)+\lambda\varphi(x,y,z)+\mu\psi(x,y,z) \\[2em]
\begin{cases}
F'_x(x,y,z,\lambda,\mu) = f'_x(x,y,z) + \lambda \varphi'_x(x,y,z) + \mu \psi'_x(x,y,z) = 0 \\[2ex]
F'_y(x,y,z,\lambda,\mu) = f'_y(x,y,z) + \lambda \varphi'_y(x,y,z) + \mu \psi'_y(x,y,z) = 0 \\[2ex]
F'_z(x,y,z,\lambda,\mu) = f'_z(x,y,z) + \lambda \varphi'_z(x,y,z) + \mu \psi'_z(x,y,z) = 0 \\[2ex]
F'_\lambda(x,y,z,\lambda,\mu) = \varphi(x,y,z) = 0 \\[2ex]
F'_\mu(x,y,z,\lambda,\mu) = \psi(x,y,z)= 0 \\
\end{cases}
</script>
</p>
<h5 id="_2">例题</h5>
<blockquote class="content-quote">
<p>函数 <script type="math/tex">f(x,y)</script> 在条件 <script type="math/tex">\phi(x,y)=0</script> 下取得极值的必要条件，设 <script type="math/tex">(x_0,y_0)</script> 处为极值点，则<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\phi(x_0,y_0) &= 0 \\[2ex]
df &= f'_x(x,y)dx + f'_y(x,y)dy  \\[2ex]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} = f'_x(x_0,y_0) + f'_y(x_0,y_0)\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} = f'_y(x_0,y_0) + f'_x(x_0,y_0)\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_x(x_0,y_0)}{\phi'_y(x_0,y_0)} \\[2ex]
\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_y(x_0,y_0)}{\phi'_x(x_0,y_0)}
\end{cases} \\[2em]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} =& f'_x(x_0,y_0) -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
&f'_x(x_0,y_0)-\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} =& f'_y(x_0,y_0) -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
&f'_y(x_0,y_0)-\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
\end{cases} \\[2em]
综上&\begin{cases}
f'_x(x_0,y_0)+\lambda\phi'_x(x_0,y_0) = 0 \\[2ex]
f'_y(x_0,y_0)+\lambda\phi'_y(x_0,y_0) = 0 \\[2ex]
\phi(x_0,y_0) = 0 \\[2ex]
\end{cases}
，且\lambda = -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)} = -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}
\end{split}\end{equation}
</script>
<br />
恰好与下式相等</p>
</blockquote>
<h2 id="15">第15讲 微分方程</h2>
<blockquote class="content-quote">
<p>微分方程的通解：若微分方程的解中含有任意常数，且任意常数的 <strong>个数</strong> 与微分方程的 <strong>阶数</strong> <strong>相同</strong>，则称之为微分方程的通解。</p>
</blockquote>
<h3 id="1_17">1. 一阶微分方程的求解</h3>
<h4 id="1_18">（1）换元</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y'&=f(x)g(y) \\[3ex]
y'&=f(ax+by+c) \ \ \ \ \ \ \ \ \ \ &u=ax+by+c\\[2ex]
y'&=f(\cfrac{y}{x}) &u=\cfrac{y}{x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\[1ex]
\cfrac{1}{y}'&=f(\cfrac{x}{y}) &u=\cfrac{x}{y} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}\end{equation}
</script>
</p>
<h4 id="2_15">（2）齐次线性</h4>
<p>
<script type="math/tex; mode=display">
y'+p(x)y=0 \\[2em]
\underline{方程两边同时乘以：e^{\int{p(x)dx}}} \\[2em]
y = Ce^{-\int{p(x)dx}}
</script>
</p>
<h4 id="3_12">（3）非齐次线性</h4>
<p>
<script type="math/tex; mode=display">
y'+p(x)y=q(x)
</script>
</p>
<p>
<script type="math/tex; mode=display">
\underline{方程两边同时乘以：e^{\int{p(x)dx}}}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
e^{\int{p(x)dx}}y'+e^{\int{p(x)dx}}p(x)y &= e^{\int{p(x)dx}}q(x) \\[3ex]
                    [e^{\int{p(x)dx}}y]' &= e^{\int{p(x)dx}}q(x) \\[3ex]
                       e^{\int{p(x)dx}}y &= \int{e^{\int{p(x)dx}}q(x)}+C \\[3ex]
                                       y &= e^{-\int{p(x)dx}}[\int{e^{\int{p(x)dx}}q(x)}+C]
\end{split}\end{equation}
</script>
</p>
<h4 id="4_8">（4）伯努利方程</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y'+p(x)y &= q(x)y^n \\[2em]
\Rightarrow y^{-n}y'+p(x)y^{1-n} &= q(x) \\[2em]
设\ \ z &= y^{1-n} \\[2ex]
z_x'&= (1-n)y^{-n}y' \\[1ex]
\cfrac{1}{1-n}z_x' &= y^{-n}y' \\[2em]
\Rightarrow \cfrac{1}{1-n}z_x'+p(x)z &= q(x) \\[2em]
\Rightarrow z_x'+(1-n)p(x)z &= (1-n)q(x)
\end{split}\end{equation}
</script>
</p>
<h3 id="2_16">2. 二阶微分方程的求解</h3>
<h4 id="1_19">（1）齐次线性方程的通解</h4>
<p>
<script type="math/tex; mode=display">
y''+py'+qy=0
</script>
</p>
<p>若 <script type="math/tex"> p^2 - 4q \gt 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个不等实根，即 <script type="math/tex"> \lambda_1\neq\lambda_2 </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = C_1e^{\lambda_1x} + C_2e^{\lambda_2x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q = 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个相等实根，即二重根，令 <script type="math/tex"> \lambda_1=\lambda_2=\lambda </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = (C_1 + C_2x) e^{\lambda x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q \lt 0 </script> ，设 <script type="math/tex"> \alpha\pm\beta i </script> 是特征方程的一对共轭复根，可得其通解为<br />
<script type="math/tex; mode=display">
y =e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)
</script>
</p>
<h4 id="2-ypyqyfx">（2）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f(x) </script>  的通解</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\big\{y''+py'+qy=f(x)的通解\big\}
= \ \ &\big\{y''+py'+qy=0的通解\big\} \\
+ &\big\{y''+py'+qy=f(x)的特解\big\}
\end{split}\end{equation}
</script>
</p>
<p>求特解的方法：</p>
<p>当 <script type="math/tex">f(x) = P_n(x)</script> 时，设特解为 <script type="math/tex">y^* = Q_n(x)</script>
</p>
<ul>
<li>
<p>比如，<script type="math/tex">P_n(x)=x</script>，则 <script type="math/tex">Q_n(x)=Ax+B</script>
</p>
</li>
<li>
<p>比如，<script type="math/tex">P_n(x)=x^2</script>，则 <script type="math/tex">Q_n(x)=Ax^2+Bx+C</script>
</p>
</li>
</ul>
<p>当 <script type="math/tex"> f(x) = e^{ax}P_n(x) </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}Q_n(x)x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
Q_n(x)\text{为x的n次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\neq\lambda_1,\alpha\ne\lambda_2\\[2ex]
1,\ \alpha=\lambda_1\text{或}\alpha=\lambda_2\\[2ex]
2,\ \alpha=\lambda_1=\lambda_2
\end{cases}
\end{cases}
</script>
</p>
<p>当 <script type="math/tex"> f(x) = e^{ax}[P_m(x)\cos\beta x + P_n(x)\sin\beta x] </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}[Q_l^{(1)}(x)\cos\beta x + Q_l^{(2)}\sin\beta x]x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
l=\max\{m,n\} \\[2ex] 
Q_l^{(1)},Q_l^{(2)}\text{为x的两个不同的l次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\pm\beta \text{i不是特征根} \\[2ex]
1,\ \alpha\pm\beta \text{i是特征根}
\end{cases}
\end{cases}
</script>
</p>
<h4 id="3-ypyqyf_1xf_2x">（3）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f_1(x)+f_2(x)</script>  的通解</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\big\{y''+py'+qy=f_1(x)+f_2(x)的通解\big\}
= \ \ &\big\{y''+py'+qy=0的通解\big\} \\
+ &\big\{y''+py'+qy=f_1(x)的特解\big\} \\
+ &\big\{y''+py'+qy=f_2(x)的特解\big\}
\end{split}\end{equation}
</script>
</p>
<h3 id="3_13">3. 应用</h3>
<h4 id="1_20">（1）曲线切线的斜率</h4>
<p>
<script type="math/tex; mode=display">
f'(x)\bigg|_{x=x_0}=\tan\alpha
</script>
</p>
<h4 id="2_17">（2）面积</h4>
<p>
<script type="math/tex; mode=display">
S = \int_a^b f(x)dx
</script>
</p>
<h4 id="3_14">（3）弧长</h4>
<p>
<script type="math/tex; mode=display">
L = \int_a^b\sqrt{1+(f_x')^2}dx
</script>
</p>
<h4 id="4-x">（4）绕 <script type="math/tex">x</script> 轴旋转体的侧面积</h4>
<p>
<script type="math/tex; mode=display">
S_{绕x轴旋转体侧} = 2\pi r·h = 2\pi\int_a^b\bigg|f(x)\bigg|\ \sqrt{1+(f_x')^2}\ dx
</script>
</p>
<h4 id="5-x">（5）绕 <script type="math/tex">x</script> 轴旋转体的体积</h4>
<p>
<script type="math/tex; mode=display">
V_x = \pi r^2 = \pi\int_a^b f^2(x)dx
</script>
</p>
<h4 id="6-y">（6）绕 <script type="math/tex">y</script> 轴旋转体的体积</h4>
<p>
<script type="math/tex; mode=display">
V_y = 2\pi r·h = 2\pi\int_a^b x\bigg|f(x)\bigg|dx
</script>
</p>
<h4 id="7_1">（7）平均值</h4>
<p>
<script type="math/tex; mode=display">
\overline f = \cfrac{1}{b-a} \int_a^b f(x)dx = f(\xi)
</script>
</p>
<h4 id="8_1">（8）曲率</h4>
<p>
<script type="math/tex; mode=display">
k=\cfrac{|f''|}{[1+(f')^2]^{\frac{3}{2}}}
</script>
</p>
<h2 id="16">第16讲 无穷级数</h2>
<h3 id="1_21">1. 数项级数</h3>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n
</script>
</p>
<p>判敛法<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\sum_{n=1}^\infty u_n收敛⟺&\lim_{n\to\infty}S_n存在⟺\{S_n\}有界 \\[1ex]
                        ⟹&\lim_{n\to\infty}u_n=0 \\[1ex]
\sum_{n=1}^\infty(u_{n+1}-u_n)收敛⟺&\lim_{n\to\infty}u_n存在
\end{split}\end{equation}
</script>
</p>
<h4 id="1_22">（1）正向级数</h4>
<h5 id="1_23">1、比较判别法</h5>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n,\ (u_n > 0)
</script>
</p>
<p>比较判别法，<strong>比较：设两个正项级数</strong><br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n,\ (u_n > 0)，\sum_{n=1}^\infty v_n,\ (v_n > 0)
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&u_n \le v_n \\[2ex]
若\sum_{n=1}^\infty v_n 收敛，&则\sum_{n=1}^\infty u_n 收敛 \\[1ex]
&若\sum_{n=1}^\infty u_n 发散，&则\sum_{n=1}^\infty v_n 发散
\end{split}\end{equation}
</script>
</p>
<p>比较判别法的极限形式<br />
<script type="math/tex; mode=display">
\lim_{n\to\infty}\cfrac{u_n}{v_n}=l
\begin{cases}
0,                 & u_n 是 v_n 高阶无穷小
\begin{cases}
\sum_{n=1}^\infty v_n收敛，&\sum_{n=1}^\infty u_n收敛 \\[1ex]
&\sum_{n=1}^\infty u_n发散，\sum_{n=1}^\infty v_n发散
\end{cases} \\[2ex]
0\lt c\lt +\infty, & u_n 是 v_n 同阶无穷小，\sum_{n=1}^\infty u_n，\sum_{n=1}^\infty v_n同敛散\\[2ex]
+\infty            & u_n 是 v_n 低阶无穷小
\begin{cases}
\sum_{n=1}^\infty v_n发散，&\sum_{n=1}^\infty u_n发散 \\[1ex]
&\sum_{n=1}^\infty u_n收敛，\sum_{n=1}^\infty v_n收敛
\end{cases}
\end{cases}
</script>
</p>
<blockquote class="content-quote">
<p>等比级数<br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty aq^{n-1}= \lim_{n\to\infty}\cfrac{a(1-q^n)}{1-q}
\begin{cases}
=\cfrac{a}{1-q}, & |q|\lt1 \\[2ex]
\text{发散}, & |q|\ge1
\end{cases}
</script>
</p>
<p>
<script type="math/tex">p</script> 级数<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
p\ 级数：
\sum_{n=1}^\infty \cfrac{1}{n^p}
&\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases} \\[2em]
广义\ p\ 级数：
\sum_{n=1}^\infty \cfrac{1}{n(\ln n)^p}
&\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases} \\[2em]
交错\ p\ 级数：
\sum_{n=1}^\infty (-1)^{n-1}\cfrac{1}{n^p}
&\begin{cases}
\text{绝对收敛}, & q\gt1 \\[2ex]
\text{条件收敛}, & 【\ 0\lt q\le1\ 】
\end{cases}
\end{split}\end{equation}
</script>
</p>
<p>反常积分<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases} \\[2em]
广义\ p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^\alpha(\ln x)^\beta}dx
&\begin{cases}
\text{收敛}, &\alpha\gt1\ 或\ \alpha=1,\beta\gt1 \\[2ex]
\text{发散}, &\alpha\lt1\ 或\ \alpha=1,\beta\le1 \\[2ex]
\end{cases} \\[2em]
瑕积分：
\int_0^1\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & 【\ q\lt1\ 】 \\[2ex]
\text{发散}, & 【\ q\ge1\ 】
\end{cases}
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h5 id="2_18">2、比值判别法（达朗贝尔）</h5>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty}\cfrac{u_{n+1}}{u_n}=\rho
\begin{cases}
\text{收敛}, & q\lt1 \\[2ex]
\text{发散}, & q\gt1 \\[2ex]
\text{失效}, & q=1
\end{cases}
</script>
</p>
<h5 id="3_15">3、根植判别式（柯西）</h5>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty}\sqrt[n]{u_n}=\rho
\begin{cases}
\text{收敛}, & q\lt1 \\[2ex]
\text{发散}, & q\gt1 \\[2ex]
\text{失效}, & q=1
\end{cases}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
比值判别法成立 &\Rightarrow 根值判别法成立 \\[2ex]
&\nLeftarrow
\end{split}\end{equation}
</script>
</p>
<h4 id="2_19">（2）交错级数</h4>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty (-1)^{n-1}u_n,\ (u_n > 0)
</script>
</p>
<p>
<script type="math/tex; mode=display">
莱布尼兹判别法：\lim_{n\to\infty} u_n = 0\ 且\ u_n \ge u_{n+1}\Rightarrow 级数收敛
</script>
</p>
<h4 id="3_16">（3）任意级数</h4>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{n\to\infty}|u_n| 收敛                           & \Rightarrow \lim_{n\to\infty}u_n 绝对收敛 \\[1em]
\lim_{n\to\infty}|u_n| 发散，\lim_{n\to\infty}u_n 收敛 & \Rightarrow \lim_{n\to\infty}u_n 条件收敛
\end{split}\end{equation}
</script>
</p>
<p>绝对收敛级数是收敛的，但收敛的级数不一定是绝对收敛级数。</p>
<h4 id="4_9">（4）条件收敛</h4>
<p>
<script type="math/tex; mode=display">
\sum u_n 收敛，\sum |u_n|发散，则该级数条件收敛
</script>
</p>
<h3 id="2_20">2. 幂级数</h3>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n \ \ 的幂级数形式：\sum_{n=0}^\infty a_n x^n
</script>
</p>
<h4 id="1_24">（1）收敛半径、收敛区间、收敛域的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&对于标准幂级数：\sum_{n=0}^\infty a_n x^n \\[2ex]
&收敛半径：R \\[2ex]
&收敛区间：(-R,R)\\[2ex]
&收敛域：(-R,R)+收敛区间端点的收敛性
\end{split}\end{equation}
</script>
</p>
<h4 id="2_21">（2）收敛半径</h4>
<h5 id="1-a_n-neq-0n012">1、不缺项幂级数 <script type="math/tex">a_n \neq 0,(n=0,1,2...)</script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{n\to\infty}\bigg|\cfrac{a_{n+1}}{a_n}\bigg| &= \rho \\[2ex]
\lim_{n\to\infty}\sqrt[n]{a_n} &= \rho \\[2em]
\sum_{n=0}^{\infty}a_nx^n \ \ 的收敛半径R &= 
\begin{cases}
\cfrac{1}{\rho}, & \rho \ne 0, +\infty\\[2ex]
+\infty,         & \rho = 0 \\[2ex]
0,               & \rho = +\infty
\end{cases}
\end{split}\end{equation}
</script>
</p>
<h5 id="2_22">2、缺项幂级数的收敛半径</h5>
<p>回归数项级数收敛域定义 <br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{n\to\infty}\cfrac{|u_{n+1}(x)|}{|u_n(x)|} &= \rho \\[2ex]
\lim_{n\to\infty}\sqrt[n]{|u_n(x)|} &= \rho \\[2ex]
\end{split}\end{equation}
</script>
</p>
<h4 id="3_17">（3）阿贝尔定理</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&幂级数\sum_{n=0}^\infty a_nx^n在点 x=x_1(x_1\neq0)处收敛，对于|x|<|x_1|的x，幂级数绝对收敛 \\
&幂级数\sum_{n=0}^\infty a_nx^n在点 x=x_2(x_2\neq0)处发散，对于|x|>|x_2|的x，幂级数发散
\end{split}\end{equation}
</script>
</p>
<h3 id="3_18">3. 展开</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
函数展开：f(x) &= \sum a_n x^n \\[1em]
积分展开：\int_a^b f(x)dx &= \sum a_n\cfrac{b^{n+1}-a^{n+1}}{n+1}\\[1em]
导数展开：\cfrac{df(x)}{dx} &= \sum na_n x^{n-1}
\end{split}\end{equation}
</script>
</p>
<h3 id="4-sx">4. 求和函数 <script type="math/tex">S(x)</script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
S(x)=\sum_{n=1}^\infty u_n(x)
\end{split}\end{equation}
</script>
</p>
<h5 id="1_25">1、直接计算或带入泰勒级数</h5>
<h5 id="2_23">2、用先积后导或先导后积的方法</h5>
<ol>
<li>先积后导</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\ \ \ \ \ \ S(x)&=\sum (an+b)x^{an} \\[2ex]
&=x\sum (an+b)x^{an-1} \\[2ex]
\int_0^x S_{tmp}(t)dt&=\int_0^x \sum (an+b)t^{an-1}dt \\[2ex]
&=\sum\int_0^x (an+b)t^{an-1}dt \\[2ex]
&=\sum (x^{an}+\cfrac{b}{an}x^{an}) \\[2ex]
&=\sum (x^a)^n+\cfrac{b}{a}\sum\cfrac{(x^a)^n}{n} \\[2ex]
S_{tmp}(x)&=(...)'\\[2ex]
S(x)&=xS_{tmp}(x)
\end{split}\end{equation}
</script>
</p>
<ol start="2">
<li>先导后积</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\ \ \ \ \ \ S(x)&=\sum \cfrac{x^{an}}{an+b} \\[2ex]
&=x^{-b}\sum \cfrac{1}{an+b}x^{an+b}\\[2ex]
S'(x)
&=-bx^{-b-1}\sum \cfrac{1}{an+b}x^{an+b}+x^{-b}\sum x^{an+b-1} \\[2ex]
&=-bx^{-1}\sum \cfrac{1}{an+b}x^{an}+\sum x^{an-1} \\[2ex]
&=-bx^{-1}S(x)+\sum x^{an-1} \\[3ex]
(若\ b=0)\ \ \ \ \ \int_0^xS(t)dt&=\int_0^x... \\[2ex]
&=S(x)-S(0) \\[2ex]
(其他, 解微分方程) \ \ \ \ ...
\end{split}\end{equation}
</script>
</p>
<ol start="3">
<li>拆分</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\ \ \ \ \ \ S(x)&=\sum \cfrac{cn^2+dn+e}{an+b}x^{an}=\sum_{(1)}+\sum_{(2)}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>例题：【2020第17题】设数列 <script type="math/tex">\{a_n\}</script> 满足 <script type="math/tex">a_1=1,(n+1)a_{n+1}=(n+\cfrac{1}{2})a_n</script>，证明：当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 收敛，并求其和函数 <script type="math/tex">S(x)</script>
</p>
<p>1、证明，当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 收敛<br />
<script type="math/tex; mode=display">
\cfrac{a_{n+1}}{a_n}=\cfrac{n+\cfrac{1}{2}}{n+1} < 1
</script>
<br />
又 <script type="math/tex">a_1=1</script>，则数列 <script type="math/tex">\{a_n\}</script> 单调递减，且 <script type="math/tex">0<a_n<1 </script>，<script type="math/tex">|a_nx^n|<|x^n|</script>
</p>
<p>当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty x^n</script> 绝对收敛，故 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 绝对收敛（绝对收敛 <script type="math/tex">\to</script> 收敛）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
S(x)&=\sum_{n=1}^\infty a_nx^n \\[1em]
S'(x)&=(\sum_{n=1}^\infty a_nx^n)'=\sum_{n=1}^\infty na_nx^{n-1} \\
&=\sum_{n=0}^\infty (n+1)a_{n+1}x^{n} \\
&=a_1+\sum_{n=1}^\infty (n+1)a_{n+1}x^{n} \\
&=a_1+\sum_{n=1}^\infty (n+\cfrac{1}{2})a_{n}x^{n} \\
&=a_1+\sum_{n=1}^\infty na_{n}x^{n}+\cfrac{1}{2}\sum_{n=1}^\infty a_{n}x^{n} \\
&=a_1+x\sum_{n=1}^\infty na_{n}x^{n-1}+\cfrac{1}{2}\sum_{n=1}^\infty a_{n}x^{n} \\
&=a_1+xS'(x)+\cfrac{1}{2}S(x) \\[1em]
\end{split}\end{equation}
</script>
<br />
解微分方程：<script type="math/tex">(1-x)S'(x)-\cfrac{1}{2}S(x)=1</script>，即 <script type="math/tex">S'(x)-\cfrac{1}{2(1-x)}S(x)=\cfrac{1}{1-x}</script>
<br />
<script type="math/tex; mode=display">
p(x)=-\cfrac{1}{2(1-x)}， q(x)=\cfrac{1}{1-x}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
S(x)&=e^{-\int p(x)dx}(\int e^{\int p(x)dx}q(x)dx+C) \\
&=e^{-\int (-\frac{1}{2(1-x)})dx}(\int e^{\int (-\frac{1}{2(1-x)})dx}(\frac{1}{1-x})dx+C) \\
&=e^{-\frac{1}{2}\ln(1-x)}(\int e^{\frac{1}{2}\ln(1-x)}(\frac{1}{1-x})dx+C) \\
&=(1-x)^{-\frac{1}{2}}(\int(1-x)^{-\frac{1}{2}}dx+C) \\
&=(1-x)^{-\frac{1}{2}}(-2(1-x)^{\frac{1}{2}}+C) \\
&=-2+\cfrac{C}{\sqrt{1-x}}\\[1em]
由于S(0)&=0，得到C=2 \\[1em]
S(x)&=-2+\frac{2}{\sqrt{1-x}}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h2 id="17">第17讲 空间几何</h2>
<h3 id="1_26">1. 向量基础知识</h3>
<h4 id="1_27">（1）数量积（内积，点乘）</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} \cdot \overrightarrow{\pmb{b}} 
= \bigg|\overrightarrow{\pmb{a}}\bigg|\cdot\bigg|\overrightarrow{\pmb{b}}\bigg|\cos\theta
</script>
</p>
<h4 id="2_24">（2）向量积（外积，叉乘）</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} \times \overrightarrow{\pmb{b}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{vmatrix}
</script>
</p>
<h4 id="3_19">（3）混合积</h4>
<p>
<script type="math/tex; mode=display">
[\pmb{a}\pmb{b}\pmb{c}] =
\pmb{a} \times \pmb{b} · \pmb{c} = 
\begin{vmatrix}
a_x & a_y & a_z \\
b_x & b_y & b_z \\
c_x & c_y & c_z
\end{vmatrix}
</script>
</p>
<h4 id="4_10">（4）方向角</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} 与\ x,y,z\ 轴的夹角\ \alpha,\beta,\gamma
</script>
</p>
<h4 id="5_6">（5）方向余弦</h4>
<p>
<script type="math/tex; mode=display">
\cos\alpha = \cfrac{a_x}{|\overrightarrow{\pmb{a}}|}, \ \ \ 
\cos\beta  = \cfrac{a_y}{|\overrightarrow{\pmb{a}}|}, \ \ \
\cos\gamma = \cfrac{a_z}{|\overrightarrow{\pmb{a}}|}
</script>
</p>
<h4 id="6_4">（6）方向单位向量</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}^o} = \cfrac{\overrightarrow{\pmb{a}}}{|\overrightarrow{\pmb{a}}|} = (\cos\alpha,\cos\beta,\cos\gamma)
</script>
</p>
<h3 id="2_25">2. 空间直线与空间平面</h3>
<h4 id="1_28">（1）空间直线的方向向量与法平面</h4>
<h5 id="1_29">1、一般式</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
A_1 x + B_1 y + C_1 z + D_1 = 0 \\[2ex]
A_2 x + B_2 y + C_2 z + D_2 = 0
\end{cases}
</script>
</p>
<h5 id="2_26">2、标准式（点向式，对称式）</h5>
<p>
<script type="math/tex; mode=display">
直线过定点：P(x_0,y_0,z_0) \\[1ex]
直线的方向向量：\overrightarrow{l} = (m,n,p) \\[2ex]
\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} \\[1em]
(m,n,p的值为0，意味着对应的分子为0)
</script>
</p>
<h5 id="3_20">3、参数式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} = t \\[1em]
\begin{cases}
x = x_0 + mt \\[2ex]
y = y_0 + nt \\[2ex]
z = z_0 + pt
\end{cases}
</script>
</p>
<h5 id="4_11">4、两点式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x-x_1}{x_2-x_1} = \cfrac{y-y_1}{y_2-y_1} = \cfrac{z-z_1}{z_2-z_1}
</script>
</p>
<h4 id="2_27">（2）空间平面与法向量</h4>
<h5 id="1_30">1、一般式方程</h5>
<p>
<script type="math/tex; mode=display">
A x + B y + C z + D = 0
</script>
</p>
<h5 id="2_28">2、点法式方程</h5>
<p>
<script type="math/tex; mode=display">
平面过定点：P(x_0,y_0,z_0) \\[1ex]
平面法向量：\overrightarrow{n} = (A,B,C) \\[2ex]
A(x-x_0) + B(y-y_0) + C(z-z_0) = 0
</script>
</p>
<h5 id="3_21">3、三点式</h5>
<p>
<script type="math/tex; mode=display">
\begin{vmatrix}
x-x_0 & y-y_0 & z-z_0 \\
x-x_1 & y-y_1 & z-z_1 \\
x-x_2 & y-y_2 & z-z_2
\end{vmatrix} = 0
</script>
</p>
<h5 id="4_12">4、截距式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x}{a}+\cfrac{y}{b}+\cfrac{z}{c} = 1
</script>
</p>
<h4 id="3_22">（3）平面与直线的关系</h4>
<h5 id="_3">直线之间的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
l_1直线方向向量：\overrightarrow{l_1} = (m_1,n_1,p_1) \\[1ex]
l_2直线方向向量：\overrightarrow{l_2} = (m_2,n_2,p_2) \\[3ex]
\cos\theta = \cfrac{\bigg|\overrightarrow{l_1}·\overrightarrow{l_2}\bigg|}{\bigg|\overrightarrow{l_1}\bigg|\bigg|\overrightarrow{l_2}\bigg|}
</script>
</p>
<h4 id="4_13">（4）平面与平面的关系</h4>
<p>
<script type="math/tex; mode=display">
\pi_1：A_1 x + B_1 y + C_1 z + D_1 = 0 \\[1ex]
\pi_2：A_2 x + B_2 y + C_2 z + D_2 = 0
</script>
</p>
<h5 id="1_31">1、平面法向量间的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
\pi_1平面法向量：\overrightarrow{n_1} = (A_1,B_1,C_1) \\[1ex]
\pi_2平面法向量：\overrightarrow{n_2} = (A_2,B_2,C_2) \\[3ex]
\cos\theta = \cfrac{|\overrightarrow{n_1}·\overrightarrow{n_2}|}{|\overrightarrow{n_1}||\overrightarrow{n_2}|}
</script>
</p>
<h5 id="2-px_0y_0z_0">2、点 <script type="math/tex">P(x_0,y_0,z_0)</script> 到平面的距离</h5>
<p>
<script type="math/tex; mode=display">
d = \cfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}
</script>
</p>
<h5 id="3_23">3、任意两个 <strong>平行</strong> 平面之间的距离</h5>
<p>
<script type="math/tex; mode=display">
d = \cfrac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}
</script>
</p>
<h4 id="5_7">（5）直线与平面的关系</h4>
<h5 id="1_32">1、直线方向向量与平面法向量的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
l直线方向向量：\overrightarrow{l} = (m,n,p) \\[1ex]
\pi平面法向量：\overrightarrow{n} = (A,B,C) \\[3ex]
\sin\theta = \cfrac{\bigg|\overrightarrow{l}·\overrightarrow{n}\bigg|}{\bigg|\overrightarrow{l}\bigg|\bigg|\overrightarrow{n}\bigg|}
</script>
</p>
<h5 id="2-l">2、过直线 <script type="math/tex">l</script> 的平面束方程</h5>
<p>
<script type="math/tex; mode=display">
直线\ l\ 由
\begin{cases}
A_1 x + B_1 y + C_1 z + D_1 = 0 \\[2ex]
A_2 x + B_2 y + C_2 z + D_2 = 0
\end{cases}
确定
</script>
</p>
<p>平面束方程写法1<br />
<script type="math/tex; mode=display">
过\ l\ 的平面束：(A_1 x + B_1 y + C_1 z + D_1) + \lambda(A_2 x + B_2 y + C_2 z + D_2) = 0 \\[1ex]
不包含平面：A_2 x + B_2 y + C_2 z + D_2 = 0
</script>
<br />
平面束方程写法2<br />
<script type="math/tex; mode=display">
过\ l\ 的平面束：\lambda_1(A_1 x + B_1 y + C_1 z + D_1) + \lambda_2(A_2 x + B_2 y + C_2 z + D_2) = 0 \\[1ex]
\lambda_1^2+\lambda_2^2 \neq 0 \\[1ex]
包含所有平面
</script>
</p>
<h3 id="3_24">3. 空间曲线和空间曲面</h3>
<h4 id="1_33">（1）空间曲线的切线与法平面</h4>
<h5 id="1_34">1、参数方程给出曲线</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
x = x(t) \\[2ex]
y = y(t) \\[2ex]
z = z(t) 
\end{cases}
</script>
</p>
<p>曲线在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的切向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} = (x'(t_0), y'(t_0), z'(t_0))
</script>
</p>
<h5 id="2_29">2、用方程组给出曲线</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}
确定
\begin{cases}
x = x \\[2ex]
y = y(x) \\[2ex]
z = z(x) 
\end{cases}
</script>
</p>
<p>曲线在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的切向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} = (1, y'(x_0), z'(x_0))
</script>
</p>
<h4 id="2_30">（2）空间曲面的切平面与法线</h4>
<h5 id="1_35">1、隐式给出曲面</h5>
<p>
<script type="math/tex; mode=display">
F(x,y,z)=0
</script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = (F'_x|_{P_0}, F'_y|_{P_0}, F'_z|_{P_0})
</script>
</p>
<h5 id="2_31">2、显式给出曲面</h5>
<p>
<script type="math/tex; mode=display">
z=z(x,y)
</script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = (z'_x(x_0,y_0), z'_y(x_0,y_0), -1)
</script>
</p>
<h5 id="3_25">3、用参数方程给出曲面</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
x = x(u,v) \\[2ex]
y = y(u,v) \\[2ex]
z = z(u,v) 
\end{cases}
</script>
</p>
<p>固定 <script type="math/tex">v=v_0</script> ，得到 <script type="math/tex">u </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{\pmb{l}_1} = (x'_u, y'_u, z'_u)|_{P_0} </script>
</p>
<p>固定 <script type="math/tex">u=u_0</script> ，得到 <script type="math/tex"> v </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{\pmb{l}_2} = (x'_v, y'_v, z'_v)|_{P_0} </script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = \overrightarrow{\pmb{l_1}} \times \overrightarrow{\pmb{l_2}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
l_{1x} & l_{1y} & l_{1z} \\
l_{2x} & l_{2y} & l_{2z}
\end{vmatrix}
</script>
</p>
<blockquote class="content-quote">
<p>曲线 <script type="math/tex">\rightarrow</script> 切向量 <script type="math/tex">\rightarrow</script> 切线 <script type="math/tex">\rightarrow</script> 法平面</p>
<p>曲面 <script type="math/tex">\rightarrow</script> 法向量 <script type="math/tex">\rightarrow</script> 法线 <script type="math/tex">\rightarrow</script> 切平面<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} \ \ \  or\ \ \ \overrightarrow{\pmb{n}} = (A, B, C) \\[1em]
\cfrac{x-x_0}{A} = \cfrac{y-y_0}{B} = \cfrac{z-z_0}{C} \\[1em]
A(x-x_0)+B(y-y_0)+C(z-z_0)=0
</script>
</p>
</blockquote>
<h3 id="4_14">4. 空间曲线在坐标面上的投影</h3>
<p>
<script type="math/tex; mode=display">
曲线\ \Gamma：
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}
在\ xOy\ 平面上的投影包含于曲线
\begin{cases}
\varphi(x,y)=0 \\[2ex]
z=0
\end{cases}
</script>
</p>
<h3 id="5_8">5. 旋转曲面</h3>
<h4 id="1_36">（1）曲线绕任意直线旋转</h4>
<p>
<script type="math/tex; mode=display">
曲线\ \Gamma：
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}\ 
绕直线\ 
L：\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} \ 
旋转
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/IMG_EFCEA317AD4E-1.jpeg" alt="IMG_EFCEA317AD4E-1" style="zoom:25%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
\overrightarrow{M_1P}\ \bot\ \overrightarrow{\pmb{s}} \\[2ex]
\bigg|\overrightarrow{M_0P}\bigg| = \bigg|\overrightarrow{M_0M_1}\bigg|
\end{cases} \\[2ex]
\Rightarrow
&\begin{cases}
m(x-x_1)+n(y-y_1)+p(z-z_1) = 0 \\[2ex]
(x-x_0)^2+(y-y_0)^2+(z-z_0)^2 = (x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2
\end{cases} \\[2ex]
且
&\begin{cases}
F(x_1,y_1,z_1)=0 \\[2ex]
G(x_1,y_1,z_1)=0
\end{cases} \\[2ex]
&消去\ x_1,y_1,z_1\ 后得到\ x,y,z\ 的表达式
\end{split}\end{equation}
</script>
</p>
<h4 id="2_32">（2）曲线绕坐标轴旋转</h4>
<p>解法1<br />
<script type="math/tex; mode=display">
绕\ x\ 轴：x=x，y=z=\pm\sqrt{y^2+z^2} \\[1ex]
绕\ y\ 轴：y=y，z=x=\pm\sqrt{z^2+x^2} \\[1ex]
绕\ z\ 轴：z=z，x=y=\pm\sqrt{x^2+y^2} \\[1ex]
绕哪个轴，哪个轴不变，其他两个轴等于另外两个轴各自的平方之和再开根号
</script>
<br />
解法2<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
F(x_1,y_1,z_1)=0 \\[2ex]
G(x_1,y_1,z_1)=0 \\[2ex]
x^2+y^2=x_1^2+y_1^2
\end{cases} \\[2ex]
&消去\ x_1,y_1,z_1\ 后得到\ x,y,z\ 的表达式
\end{split}\end{equation}
</script>
</p>
<h3 id="6_5">6. 场论</h3>
<h4 id="1_37">（1）方向导数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial\overrightarrow{\pmb{l}}} \bigg|_{P_0}
&= \lim_{t \to 0^+}\frac{u(x_0 + \Delta x, y_0 + \Delta y, z_0 + \Delta z) - u(x_0, y_0, z_0)}{t} \\[2ex]
&= \lim_{t \to 0^+}\frac{u(x_0 + t\cos\alpha, y_0 + t\cos\beta, z_0 + t\cos\gamma) - u(x_0, y_0, z_0)}{t} \\[3ex]
(若u可微)
&= \lim_{t \to 0^+}\frac{u'_x(P_0)\Delta x + u'_y(P_0)\Delta y + u'_z(P_0)\Delta z - o(t)}{t} \\[3ex]
&= u'_x(P_0)\cos\alpha + u'_y(P_0)\cos\beta + u'_z(P_0)\cos\gamma \\[3em]
其中\ \overrightarrow{\pmb{l}^o}&=(\cos\alpha,\cos\beta,\cos\gamma)\ 为单位向量 \\[2ex]
t &= \sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}\\[3em]
\end{split}\end{equation}
</script>
</p>
<h5 id="_4">方向导数与偏导数的关系</h5>
<p>偏导数的定义：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial x}
&= \lim_{\Delta x \to 0}\frac{u(x_0 + \Delta x, y_0, z_0) - u(x_0, y_0, z_0)}{\Delta x} = u_x' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(1,0,0)\ 即 \ x\ 轴正方向的方向导数\\[2ex]
\cfrac{\partial u}{\partial y}
&= \lim_{\Delta y \to 0}\frac{u(x_0, y_0 + \Delta y, z_0) - u(x_0, y_0, z_0)}{\Delta y} = u_y' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(0,1,0)\ 即 \ y\ 轴正方向的方向导数\\[2ex]
\cfrac{\partial u}{\partial z}
&= \lim_{\Delta z \to 0}\frac{u(x_0, y_0, z_0 + \Delta z) - u(x_0, y_0, z_0)}{\Delta z} = u_z' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(0,0,1)\ 即 \ z\ 轴正方向的方向导数\\[2ex]
\end{split}\end{equation}
</script>
</p>
<h4 id="2_33">（2）梯度</h4>
<p>若 <script type="math/tex">u(x,y,z)</script> 具有一阶偏导数<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\overrightarrow{\mathbf{grad}}\ u \bigg|_{P_0}
&= \cfrac{\partial u}{\partial x}\ \overrightarrow{\pmb{i}} + \cfrac{\partial u}{\partial y}\ \overrightarrow{\pmb{j}} + \cfrac{\partial u}{\partial z}\ \overrightarrow{\pmb{k}} \\[3ex]
&= (u'_x(P_0), u'_y(P_0), u'_z(P_0))
\end{split}\end{equation}
</script>
</p>
<h5 id="_5">方向导数与梯度的关系</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial \overrightarrow{\pmb{l}}} \bigg|_{P_0}
&= u'_x\cos\alpha + u'_y\cos\beta + u'_z\cos\gamma \\
&= (u'_x, u'_y, u'_z) · (\cos\alpha,\cos\beta,\cos\gamma) \\[2ex]
&= \overrightarrow{\mathbf{grad}}\ u \big|_{P_0}· \overrightarrow{\pmb{l}^o} \\[2ex]
&= \bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|·\bigg|\ \overrightarrow{\pmb{l}^o}\ \bigg|\cos\theta \\[2ex]
&= \bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|\cos\theta\\[1em]
\end{split}\end{equation}
</script>
</p>
<ul>
<li>由于 <script type="math/tex">\bigg|\ \overrightarrow{\pmb{l}}\ \bigg|=1</script> ，<strong>梯度的方向</strong> 即 <strong>取得最大方向导数的方向</strong></li>
<li>最大方向导数为 <script type="math/tex">\bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|</script>，即 <strong>梯度的模</strong></li>
<li>最小方向导数为 <script type="math/tex">-\bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|</script>
</li>
</ul>
<h4 id="3_26">（3）散度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{\pmb{A}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}} </script> ，则<br />
<script type="math/tex; mode=display">
div\ \overrightarrow{\pmb{A}} = \cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z}
</script>
</p>
<ul>
<li>
<p>
<script type="math/tex"> div\ \overrightarrow{\pmb{A}} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处源头的强弱程度。</p>
</li>
<li>
<p>若  <script type="math/tex"> div\ \overrightarrow{\pmb{A}} = 0 </script>  在场内处处成立，则称A为<strong>无源场</strong>。</p>
</li>
</ul>
<h4 id="4_15">（4）旋度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{\pmb{A}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}} </script> ，则<br />
<script type="math/tex; mode=display">
\overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}
</script>
</p>
<ul>
<li>
<script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处最大旋转趋势的度量。</li>
<li>若  <script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} = 0 </script>  在场内处处成立，则称A为<strong>无旋场</strong>。</li>
</ul>
<h2 id="18">第18讲 多元函数积分学</h2>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>多元函数积分学</th>
<th>表达式</th>
<th>坐标系</th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>1.二重积分</td>
<td>
<script type="math/tex">\iint_D f(x,y)d\sigma</script>
</td>
<td>直角坐标系</td>
<td>极坐标系</td>
<td></td>
</tr>
<tr>
<td>2.三重积分</td>
<td>
<script type="math/tex">\iiint_\Omega f(x,y,z)dv</script>
</td>
<td>直角坐标系</td>
<td>柱面坐标系</td>
<td>球面坐标系</td>
</tr>
<tr>
<td>3.第一型曲线积分</td>
<td>
<script type="math/tex">\int_Lf(x,y)ds</script><br /><script type="math/tex">\int_\Gamma f(x,y,z)ds</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>4.第二型曲线积分</td>
<td>
<script type="math/tex">\int_L Pdx + Qdy</script><br /><script type="math/tex">\int_\Gamma Pdx+Qdy+Rdz</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>（1）格林公式</td>
<td></td>
<td>二重积分</td>
<td></td>
<td></td>
</tr>
<tr>
<td>（2）斯托克斯公式</td>
<td></td>
<td>二重积分</td>
<td></td>
<td></td>
</tr>
<tr>
<td>5.第一型曲面积分</td>
<td>
<script type="math/tex">\iint_\Sigma f(x,y,z)dS</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>6.第二型曲面积分</td>
<td>
<script type="math/tex">\iint_\Sigma Pdydz + Qdzdx + Rdxdy</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>（1）高斯公式</td>
<td></td>
<td>三重积分</td>
<td></td>
<td></td>
</tr>
</tbody>
</table></div>
<h3 id="1_38">1. 二重积分</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(直角坐标系)\iint_D f(x,y)d\sigma &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} f(x,y)dy \\[1ex]
&= \int_{a}^{b} dy \int_{x_1(x)}^{x_2(x)} f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/IMG_0092-3451157.jpg" alt="IMG_0092" style="zoom:18%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
x = r\cos\theta \\[2ex]
y = r\sin\theta \\[2ex]
d\sigma = dxdy = r\ dr d\theta
\end{cases} \\[1em]
(极坐标系)(a)\iint_{D}f(x,y)d\sigma &= \int_\alpha^\beta d\theta\int_{r_1(\theta)}^{r_2(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D外部）}\\
(b)\iint_{D}f(x,y)d\sigma &= \int_\alpha^\beta d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D边界上）} \\
(c)\iint_{D}f(x,y)d\sigma &= \int_0^{2\pi}d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D内部）}
\end{split}\end{equation}
</script>
</p>
<h3 id="2_34">2. 三重积分</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(直角坐标系)\iiint_\Omega f(x,y,z)dv &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} dy \int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z)dz \\[1em]
&= \int_{a}^{b} dz \iint_{D_{xy}} f(x,y,z) dxdy \\[3em]
& \begin{cases}
x = r\cos\theta \\[2ex]
y = r\sin\theta \\[2ex]
z = z \\[2ex]
dv = dxdydz = r\ \ dr d\theta dz
\end{cases} \\[1em]
(柱面坐标系) &= \iiint_\Omega f(r\cos\theta,\ r\sin\theta,\ z)r\ \ dr d\theta dz \\[3em]
& \begin{cases}
x = r\sin\varphi\cos\theta \\[2ex]
y = r\sin\varphi\sin\theta \\[2ex]
z = r\cos\varphi \\[2ex]
dv = dxdydz = r^2\sin\varphi\ \ d\theta d\varphi dr
\end{cases} \\[1em]
(球面坐标系) &= \iiint_\Omega f(r\sin\varphi\cos\theta,\ r\sin\varphi\sin\theta,\ r\cos\varphi)r^2\sin\varphi\ \ d\theta d\varphi dr
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
(2009.12)
\begin{equation}\begin{split}
设\ \Omega=\{(x,y,z)&|x^2+y^2+z^2\le1\} \\[1ex]
\iiint_\Omega z^2dv &= \cfrac{1}{3}\iiint_\Omega x^2+y^2+z^2dv=\cfrac{1}{3}\iiint_\Omega dv\ \ \ \ \  错误！！！\\[1ex]
&=\cfrac{1}{3}\iiint_\Omega r^2r^2\sin\varphi d\theta d\varphi dr \\[1ex]
&=\cfrac{1}{3}\int_0^{2\pi}d\theta\int_0^{\pi}\sin\varphi\int_0^1r^4dr \\[1ex]
&=\cfrac{1}{3}·2\pi·2·\cfrac{1}{5}=\cfrac{4\pi}{15}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h3 id="3_27"><a style="color:rgb(0,141,255)">3. 第一型曲线积分</a></h3>
<p>（数量值函数在曲线上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(二维)\int_Lf(x,y)ds &= \int_\alpha^\beta f[x(t),y(t)]\sqrt{(x_t')^2+(y_t')^2}dt \\[1ex]
&= \int_\alpha^\beta f[x,y(x)]\sqrt{1+(y_x')^2}dx \\[1ex]
&= \int_\alpha^\beta f[r(\theta)\cos\theta,r(\theta)\sin\theta]\sqrt{(r_\theta)^2+(r_\theta')^2}d\theta
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(三维)\int_\Gamma f(x,y,z)ds &= \int_\alpha^\beta f[x(t),y(t),z(t)]\sqrt{(x_t')^2+(y_t')^2+(z_t')^2}dt \\[1ex]
&= \int_\alpha^\beta f[x,y(x),z(x)]\sqrt{1+(y_x')^2+(z_x')^2}dx
\end{split}\end{equation}
</script>
</p>
<p>形心<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\int_Lxds}{\int_Lds},\cfrac{\int_Lyds}{\int_Lds},\cfrac{\int_Lzds}{\int_Lds})
</script>
<br />
重心（质心）<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\int_Lx\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds},\cfrac{\int_Ly\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds},\cfrac{\int_Lz\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds})
</script>
<br />
转动惯量（对 <script type="math/tex">x,y,z</script> 轴和原点 <script type="math/tex">O</script> 的转动惯量）<br />
<script type="math/tex; mode=display">
I_x=\int_L(y^2+z^2)\rho(x,y,z)ds \\[1ex]
I_y=\int_L(z^2+x^2)\rho(x,y,z)ds \\[1ex]
I_z=\int_L(x^2+y^2)\rho(x,y,z)ds \\[1ex]
I_O=\int_L(x^2+y^2+z^2)\rho(x,y,z)ds \\
</script>
</p>
<h3 id="4_16"><a style="color:rgb(0,141,255)">4. 第二型曲线积分</a></h3>
<p>（向量值函数在曲线上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L\overrightarrow{\pmb{F}}(x,y)\overrightarrow{\pmb{ds}} 
&= \int_L Pdx + Qdy \\[1ex]
&= \int_{t_1}^{t_2} \bigg\{\ P[x(t),y(t)]x_t'(t) + Q[x(t),y(t)]y_t'(t)\ \bigg\}dt
\end{split}\end{equation}
</script>
</p>
<h4 id="_6">【二维】第一二型曲线积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L Pdx + Qdy 
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\cos\beta)ds \\[1ex]
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\sin\alpha)ds\\[2em]
方向余弦\ \cos\alpha &= \cfrac{x_t'(t)}{\sqrt{(x_t')^2+(y_t')^2}} \\[1ex]
\cos\beta &= \cfrac{y_t'(t)}{\sqrt{(x_t')^2+(y_t')^2}} = \sin\alpha \\[2ex]
\overrightarrow{\pmb{l}}=(\cos\alpha,\cos\beta)\ &为\ L\ 上点\ (x,y)\ 处与\ L\ 同向的单位切向量. \\[1ex]
=(\cos\alpha,\sin\alpha)\ 
\end{split}\end{equation}
</script>
</p>
<h4 id="_7">【二维】格林公式</h4>
<p>（第二型曲线积分与二重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_LPdx+Qdy &= \iint_D(\cfrac{\partial Q}{\partial x} - \cfrac{\partial P}{\partial y})dxdy
\end{split}\end{equation}
</script>
</p>
<p>////////////////////////////////////////////////////////<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_\Gamma\overrightarrow{\pmb{F}}(x,y)\overrightarrow{\pmb{ds}} 
&= \int_\Gamma Pdx + Qdy +Rdz \\[1ex]
&= \int_{t_1}^{t_2} \bigg\{\ P[x(t),y(t),z(t)]x_t'(t) + Q[x(t),y(t),z(t)]y_t'(t) + R[x(t),y(t),z(t)]z_t'(t)\ \bigg\}dt
\end{split}\end{equation}
</script>
</p>
<h4 id="_8">【三维】第一二型曲线积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_\Gamma Pdx+Qdy+Rdz
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\cos\beta + R\cos\gamma)ds \\[1ex]
方向余弦\ \cos\alpha &= \cfrac{x_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[1ex]
\cos\beta &= \cfrac{y_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[2ex]
\cos\gamma &= \cfrac{z_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[2ex]
\overrightarrow{\pmb{l}}=(\cos\alpha,\cos\beta,\cos\gamma)\ &为\ L\ 上点\ (x,y,z)\ 处与\ L\ 同向的单位切向量.
\end{split}\end{equation}
</script>
</p>
<h4 id="_9">【三维】斯托克斯公式</h4>
<p>（第二型曲线积分与二重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_\Gamma Pdx+Qdy+Rdz
&= \iint_D \begin{vmatrix}
dydz & dzdx & dxdy \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} &\text{(第二型曲面积分)} \\[1ex]
&= \iint_D \begin{vmatrix}
\cos\alpha & \cos\beta & \cos\gamma  \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} dS \ \ \ \ &\text{(第一型曲面积分)}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\oint_\Gamma Pdx+Qdy+Rdz = \iint_D
 (\cfrac{\partial R}{\partial y} - \cfrac{\partial Q}{\partial z})dydz
+(\cfrac{\partial P}{\partial z} - \cfrac{\partial R}{\partial x})dzdx
+(\cfrac{\partial Q}{\partial x} - \cfrac{\partial P}{\partial y})dxdy
</script>
</p>
<h3 id="5_9"><a style="color:rgb(0,141,255)">5. 第一型曲面积分</a></h3>
<p>（数量值函数在曲面上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma f(x,y,z)dS &= \iint_{D_{xy}} f(x,y,z(x,y))\sqrt{1+(z_x')^2+(z_y')^2}dxdy
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/IMG_0093-3451157.jpg" alt="IMG_0093" style="zoom: 16%;" /></p>
<p>形心<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\iint_\Sigma xdS}{\iint_\Sigma dS},\cfrac{\iint_\Sigma ydS}{\iint_\Sigma dS},\cfrac{\iint_\Sigma zdS}{\iint_\Sigma dS})
</script>
<br />
曲面重心（质心）<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\iint_\Sigma x\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS},\cfrac{\iint_\Sigma y\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS},\cfrac{\iint_\Sigma z\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS})
</script>
<br />
转动惯量（对 <script type="math/tex">x,y,z</script> 轴和原点 <script type="math/tex">O</script> 的转动惯量）<br />
<script type="math/tex; mode=display">
I_x=\iint_\Sigma(y^2+z^2)\rho(x,y,z)dS \\[1ex]
I_y=\iint_\Sigma(z^2+x^2)\rho(x,y,z)dS \\[1ex]
I_z=\iint_\Sigma(x^2+y^2)\rho(x,y,z)dS \\[1ex]
I_O=\iint_\Sigma(x^2+y^2+z^2)\rho(x,y,z)dS \\
</script>
</p>
<h3 id="6_6"><a style="color:rgb(0,141,255)">6. 第二型曲面积分</a></h3>
<p>（向量值函数在曲面上的积分）</p>
<p>第二型曲面积分的被积函数 <script type="math/tex">\overrightarrow{\pmb{F}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}}</script> 定义在光滑的空间有向曲面 <script type="math/tex">\Sigma</script> 上，其物理背景是向量函数 <script type="math/tex">\overrightarrow{\pmb{F}}(x,y,z)</script> 通过曲面 <script type="math/tex">\Sigma</script> 的通量.<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma\overrightarrow{\pmb{F}}(x,y,z)\overrightarrow{\pmb{dS}} 
&=\ \ \ \iint_\Sigma \ \ \ Pdydz + Qdzdx + Rdxdy \\[1ex]
&=\ \ \ \iint_{\Sigma} \ \ \ P(x(y,z),y,z)dydz + \iint_{\Sigma} \ \ \ Q(x,y(z,x),z)dzdx + \iint_{\Sigma} \ \ \ R(x,y,z(x,y))dxdy \\[1ex]
&=\pm \iint_{D_{yz}} P(x(y,z),y,z)dydz \pm\iint_{D_{zx}} Q(x,y(z,x),z)dzdx \pm\iint_{D_{xy}} R(x,y,z(x,y))dxdy
\end{split}\end{equation}
</script>
</p>
<ol>
<li>
<p>当 <script type="math/tex">\Sigma</script> 的法向量与 <script type="math/tex">x,y,z</script> 轴的夹角为 <strong>锐角</strong> 时， <script type="math/tex">\pm</script> 取 <script type="math/tex">+</script>
</p>
</li>
<li>
<p>当 <script type="math/tex">\Sigma</script> 的法向量与 <script type="math/tex">x,y,z</script> 轴的夹角为 <strong>钝角</strong> 时， <script type="math/tex">\pm</script> 取 <script type="math/tex">-</script>
</p>
</li>
<li>
<p>即 <script type="math/tex"> \Sigma </script> 前右上侧为 <script type="math/tex"> + </script> ，后左下侧为 <script type="math/tex"> - </script>
</p>
</li>
</ol>
<h4 id="_10">第一二型曲面积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&(第一型曲面积分) \\[1ex]
\iint_\Sigma\overrightarrow{\pmb{F}}(x,y,z)\overrightarrow{\pmb{dS}}
&= \iint_\Sigma Pdydz + Qdzdx + Rdxdy\\[1ex]
&= \iint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS \\[1ex]
&= \iint_\Sigma (P\cfrac{\cos\alpha}{\cos\gamma} + Q\cfrac{\cos\beta}{\cos\gamma} + R)\cos\gamma \ dS \\[1ex]
&= \iint_\Sigma (P(-z_x') + Q(-z_y') + R)dS \\[3em]
方向余弦\ \cos\alpha &= \cfrac{-z_x'}{\sqrt{1+(z_x')^2+(z_y')^2}} \\[1ex]
\cos\beta &= \cfrac{-z_y'}{\sqrt{1+(z_x')^2+(z_y')^2}} \\[1ex]
\cos\gamma &= \cfrac{1}{\sqrt{1+(z_x')^2+(z_y')^2}}\end{split}\end{equation}
</script>
</p>
<h4 id="_11">高斯公式</h4>
<p>（第二型曲面积分与三重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iiint_\Omega(\cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z})dv &= \oiint_\Sigma Pdydz+Qdzdx+Rdxdy \\[1ex]
&= \oiint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS
\end{split}\end{equation}
</script>
</p>
<h3 id="7_2">7. 实际应用</h3>
<p>在求解时应该充分利用函数的对称性，如奇对称、偶对称、<u>轮换对称性</u>&hellip;</p>
</div>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_3">3. 导数与微分的计算</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_3">4. 反函数求导</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5_2">5. 幂指函数求导</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6_1">6. 参数方程求导</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#7">7. 莱布尼茨公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#8">8. 可微、可导、连续、可积的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#5-7">第5-7讲 一元函数微分学的应用</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_5">1. 极值、单调性</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1-yx">（1）一元函数 <script type="math/tex">y(x)</script> 的极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2-fxy0-yx-yx">（2）一元函数隐函数 <script type="math/tex"> F(x,y)=0 </script> 确定 <script type="math/tex">y=(x)</script>，求 <script type="math/tex">y(x)</script> 的极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_4">2. 拐点、凹凸性</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_4">3. 渐近线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_6">（1）水平渐近线和铅直渐近线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2x">（2）斜渐近线的正确求法(在x趋向于无穷时)</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_4">4. 曲率与曲率半径</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5_3">5. 中值定理</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_7">定理1（费马定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_5">定理2（罗尔定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_5">定理3（拉格朗日中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_5">定理4（柯西中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5_4">定理5（泰勒中值定理）（泰勒公式 / 麦克劳林公式）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6_2">定理6（积分中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#8-9">第8-9讲 一元函数积分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_8">1. 公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_9">（1）重要的公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_6">（2）被积函数包含三角函数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_7">2. 不定积分的计算方法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_10">（1）凑微</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_8">（2）换元</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_6">（3）分部积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_7">3. 定积分的计算</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_11">（1）定积分的定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_12">1、均匀分割</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_9">2、等差分割</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_8">3、等比分割</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_10">（2）重要公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_9">（3）华式公式（“点火公式”）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_6">（4）伽马函数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_7">4. 变限积分的计算</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5_5">5. 反常积分的计算</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#10-12">第10-12讲 一元函数积分学的应用</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#13">第13讲 多元函数微分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_13">1. 导数与微分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_14">（1）偏导数的定义公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_11">（2）二元函数微分的定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_12">2. 复合函数求导法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_15">（1）链式求导法则</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_13">（2）全导数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_10">（3）全微分形式不变性</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_11">3. 隐函数求导法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_16">（1）一个方程的情形</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_14">（2）方程组情形</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4-fxy">4. 二元函数 <script type="math/tex">f(x,y)</script> 极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5-fxyz0-zxy-zxy">5. 多元函数隐函数 <script type="math/tex"> F(x,y,z)=0 </script> 确定 <script type="math/tex">z(x,y)</script>，求 <script type="math/tex">z(x,y)</script> 的极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6_3">6. 条件极值（拉格朗日数乘法）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#fxy">【二元】函数 <script type="math/tex">f(x,y)</script> 在条件</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#fxyz">【三元】函数 <script type="math/tex">f(x,y,z)</script> 在条件</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_2">例题</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#15">第15讲 微分方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_17">1. 一阶微分方程的求解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_18">（1）换元</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_15">（2）齐次线性</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_12">（3）非齐次线性</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_8">（4）伯努利方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_16">2. 二阶微分方程的求解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_19">（1）齐次线性方程的通解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2-ypyqyfx">（2）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f(x) </script>  的通解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3-ypyqyf_1xf_2x">（3）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f_1(x)+f_2(x)</script>  的通解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_13">3. 应用</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_20">（1）曲线切线的斜率</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_17">（2）面积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_14">（3）弧长</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4-x">（4）绕 <script type="math/tex">x</script> 轴旋转体的侧面积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5-x">（5）绕 <script type="math/tex">x</script> 轴旋转体的体积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6-y">（6）绕 <script type="math/tex">y</script> 轴旋转体的体积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#7_1">（7）平均值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#8_1">（8）曲率</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#16">第16讲 无穷级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_21">1. 数项级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_22">（1）正向级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_23">1、比较判别法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_18">2、比值判别法（达朗贝尔）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_15">3、根植判别式（柯西）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_19">（2）交错级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_16">（3）任意级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_9">（4）条件收敛</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_20">2. 幂级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_24">（1）收敛半径、收敛区间、收敛域的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_21">（2）收敛半径</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_22">2、缺项幂级数的收敛半径</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_17">（3）阿贝尔定理</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_18">3. 展开</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_25">1、直接计算或带入泰勒级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_23">2、用先积后导或先导后积的方法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#17">第17讲 空间几何</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_26">1. 向量基础知识</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_27">（1）数量积（内积，点乘）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_24">（2）向量积（外积，叉乘）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_19">（3）混合积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_10">（4）方向角</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5_6">（5）方向余弦</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6_4">（6）方向单位向量</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_25">2. 空间直线与空间平面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_28">（1）空间直线的方向向量与法平面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_29">1、一般式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_26">2、标准式（点向式，对称式）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_20">3、参数式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#4_11">4、两点式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_27">（2）空间平面与法向量</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_30">1、一般式方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_28">2、点法式方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_21">3、三点式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#4_12">4、截距式</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_22">（3）平面与直线的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_3">直线之间的夹角（锐角/直角）</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_13">（4）平面与平面的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_31">1、平面法向量间的夹角（锐角/直角）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2-px_0y_0z_0">2、点 <script type="math/tex">P(x_0,y_0,z_0)</script> 到平面的距离</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_23">3、任意两个 <strong>平行</strong> 平面之间的距离</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5_7">（5）直线与平面的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_32">1、直线方向向量与平面法向量的夹角（锐角/直角）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2-l">2、过直线 <script type="math/tex">l</script> 的平面束方程</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_24">3. 空间曲线和空间曲面</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_33">（1）空间曲线的切线与法平面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_34">1、参数方程给出曲线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_29">2、用方程组给出曲线</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_30">（2）空间曲面的切平面与法线</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_35">1、隐式给出曲面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_31">2、显式给出曲面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_25">3、用参数方程给出曲面</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_14">4. 空间曲线在坐标面上的投影</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5_8">5. 旋转曲面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_36">（1）曲线绕任意直线旋转</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_32">（2）曲线绕坐标轴旋转</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6_5">6. 场论</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_37">（1）方向导数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_4">方向导数与偏导数的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_33">（2）梯度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_5">方向导数与梯度的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_26">（3）散度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_15">（4）旋度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#18">第18讲 多元函数积分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_38">1. 二重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_34">2. 三重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_27"><a style="color:rgb(0,141,255)">3. 第一型曲线积分</a></a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_16"><a style="color:rgb(0,141,255)">4. 第二型曲线积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_6">【二维】第一二型曲线积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_7">【二维】格林公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_8">【三维】第一二型曲线积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_9">【三维】斯托克斯公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5_9"><a style="color:rgb(0,141,255)">5. 第一型曲面积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6_6"><a style="color:rgb(0,141,255)">6. 第二型曲面积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_10">第一二型曲面积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_11">高斯公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#7_2">7. 实际应用</a>
</li>

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